View source for Algebraic number theory

The branch of number theory with the basic aim of studying properties of algebraic integers in algebraic number fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a0116001.png" /> of finite degree over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a0116002.png" /> of rational numbers (cf. [[Algebraic number|Algebraic number]]).

The set of algebraic integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a0116003.png" /> of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a0116004.png" /> — an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a0116005.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a0116006.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a0116007.png" /> (cf. [[Extension of a field|Extension of a field]]) — can be obtained from an integral basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a0116008.png" />; this means that each algebraic integer (i.e. element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a0116009.png" />) can be written in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160010.png" /> where all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160011.png" /> run through the rational integers (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160012.png" />). Moreover, such a representation is unique for each algebraic integer in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160013.png" />.

However, properties of rational integers often do not have obvious analogues for algebraic integers. The first such property is related to units, the invertible elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160014.png" /> (cf. [[Unit|Unit]]). The field of rational numbers has only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160016.png" /> as units, but a general algebraic number field may contain an infinite number of units. E.g. consider the real quadratic field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160018.png" /> is a rational integer not equal to a square. If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160019.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160021.png" /> is an integral basis for it. The [[Pell equation|Pell equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160022.png" /> has an infinite number of solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160023.png" />. Any of them gives rise to a unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160025.png" />. In fact,

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and

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is also an algebraic integer in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160028.png" />. The units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160029.png" /> form an infinite multiplicative group (the group of Pell units). The question arises: What is the structure of this group?

A second property of rational numbers without an obvious analogue for algebraic numbers is related to the theorem on unique factorization of rational integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160030.png" /> in prime factors:

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For algebraic numbers such a unique factorization need not hold. E.g. consider the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160032.png" />. In it, the number 6 has two essentially different factorizations: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160034.png" />. For extensions of higher degree the situation becomes still more complicated. The question arises: What becomes of the unique factorization theorem, does it have a meaning at all in algebraic number fields?

A third property without an obvious analogue is related to prime numbers. An ordinary prime number need not remain a prime number in an algebraic number field. E.g. in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160035.png" /> of Gaussian numbers the prime number 5 has a factorization: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160036.png" />. However, the prime number 7 remains prime in this field. The question arises: Do there exist general laws governing the behaviour of prime numbers in algebraic number fields of higher degree? In other words, is it possible to find rules that would give a definite answer to the following question: Does a given prime number remain prime in a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160037.png" /> or does it split in it, and if it splits, in how many factors?

Finally, a last (fourth) problem concerns the general structure of algebraic number fields. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160038.png" /> is the minimal field of characteristic zero without proper subfields. Any other algebraic number field does have subfields. E.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160039.png" /> is a subfield of any algebraic number field. The question arises: How many subfields are contained in a given extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160040.png" />, finitely or infinitely many, and what is their structure?

These are four main problems in algebraic number theory, and answering them constitutes the content of algebraic number theory. It is quite natural to start with the fourth problem, since its answer will shed light on the other three. The problem was solved by E. Galois in the 1820s (cf. [[Galois theory|Galois theory]]). The fact that the number of subfields of an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160041.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160042.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160043.png" /> is finite follows from the existence of a one-to-one correspondence (the fundamental Galois correspondence) between all subfields of the normal closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160044.png" /> (cf. [[Normal extension|Normal extension]]) and the subgroups of its Galois group, a finite group of finite order (number of elements) at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160045.png" />.

The structure of the group of units of a field was elucidated by P. Dirichlet. His basic idea can be given by taking the group of Pell units (cf. above) as example. Any power of such a unit (both positive or negative) is a unit. There is a fundamental unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160046.png" />, and all other units are plus or minus integral powers of it; hence, the Pell units form the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160047.png" /> with an infinite cyclic group. This fact is a special case of the general Dirichlet unit theorem for algebraic number fields: If the degree of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160048.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160050.png" /> is the number of real imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160052.png" /> the number of complex-conjugate pairs of complex imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160053.png" />, then the group of units <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160054.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160055.png" /> is the product of a finite cyclic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160057.png" /> infinite cyclic groups:

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Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160059.png" /> are independent fundamental units and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160060.png" /> is the group of roots of unity contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160061.png" />. The norm of any unit in a field, i.e. the product of the unit by its conjugates, is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160062.png" />.

The problem of the non-unique factorization of algebraic integers in algebraic number fields was solved by E. Kummer, who started from a special case; he tried to solve Fermat's last theorem on the impossibility of solving the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160063.png" /> in non-zero integers for any prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160064.png" />. Kummer expanded the left-hand side using <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160065.png" />-th roots of unity, and hence the problem was reduced to one about the algebraic integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160067.png" /> a primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160068.png" />-th root of unity. If there would be unique factorization into prime factors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160069.png" />, then it would have been sufficient to prove that not all prime factors at the left-hand side occur with an exponent that is a multiple of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160070.png" />. This was Kummer's first point of view, but Dirichlet pointed out to him that unique factorization need not hold. In order to overcome this difficulty Kummer introduced ideal numbers, thus altering the entire structure of algebraic number theory for the future. The concept of an ideal number arises from the fact that if a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160071.png" /> does not contain prime numbers into which any algebraic integer in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160072.png" /> can be split, then there is a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160073.png" /> of finite degree over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160074.png" /> in which there does exist a collection of numbers that play the role of primes for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160075.png" />. These numbers were called ideal by Kummer (since they do not lie in the original field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160076.png" />). By introducing ideal numbers the theorem on unique factorization in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160077.png" /> holds. Two numbers in a field that differ by a unit (so-called associated numbers) have one and the same ideal factors. (Note that ideal numbers are defined relative to the original field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160078.png" />; for another field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160079.png" /> one must construct an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160080.png" /> (of possibly different degree over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160081.png" />) in which all ideal numbers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160082.png" /> are contained.)

Kummer also introduced the concept of the ideal class number of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160083.png" />. Two ideal numbers are said to belong to the same class if their quotient belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160084.png" />. The number of classes thus obtained is called the ideal class number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160085.png" />. He obtained the following important result: The class number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160086.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160087.png" /> is finite, and the classes form an Abelian group under multiplication. Thus, any ideal number can be regarded as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160088.png" />-th root of some element of the original field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160089.png" />. The class number can be explicitly described in terms of other field constants (the regulator, the discriminant and the degree of the field).

Subsequently, the concept of an ideal number was replaced by that of an [[Ideal|ideal]], equivalent to it, which can be described in terms of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160090.png" /> itself. Already in the 1950s the concept of an ideal was generalized to the more comprehensive concept of a [[Divisor|divisor]]. Therefore, the modern theory of Kummer may be stated in terms of divisors. For algebraic number fields, however, the concepts of a divisor and an ideal coincide, and such fields only will be considered in the sequel. The concept of an ideal is related to that of non-associated numbers, which helps in understanding the deep connections between Kummer's theory and Dirichlet's theory of units. Although Kummer did not succeed in solving Fermat's problem, his ideas extended far beyond this problem and the concept of an ideal has now become fundamental in many branches of mathematics.

Since prime ideal numbers are defined relative to a field or, in modern terminology, since they are prime ideals in a field, the third problem on the factorization of ordinary prime numbers in an algebraic number field can be stated in the following general form. Suppose one is given a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160091.png" /> and a prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160092.png" /> of its ring of algebraic integers. The question is: Does <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160093.png" /> remain prime in an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160094.png" /> or does it split in a product of prime ideals of the ring of algebraic integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160095.png" />? In the latter case: By which law does it split? This question led to class field theory, a central part of modern algebraic number theory. The first solution of this problem was given by Kummer, who proved that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160096.png" /> is a root of an irreducible polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160098.png" /> is an integral basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a01160099.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600100.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600101.png" /> splits in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600102.png" />  "by the same law as fx"  does in the residue field (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600103.png" />). In other words, the factorization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600104.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600105.png" /> is determined by the congruence

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600106.png" /></td> </tr></table>

The corresponding factorization is called Kummer's formula (or Kummer's decomposition)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600107.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600108.png" /> are prime ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600109.png" />.

In principle, this equation solves the third problem in algebraic number theory, but it is a local equation in the sense that it is necessary to check it for each prime ideal separately. The problem of partitioning the set of all prime ideals in classes such that the factorization law is the same in any given class and such that, moreover, it is possible to give a simple description of these classes, is solved by class field theory for extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600110.png" /> with Abelian Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600111.png" />.

Equation (1) yields a preliminary concept of a class. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600112.png" /> be the degree of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600113.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600114.png" /> be the relative degree of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600115.png" />. Computing the relative norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600116.png" /> of both parts of (1) leads to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600117.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600118.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600119.png" /> are natural numbers. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600120.png" /> fixed equation (2) has a finite number of solutions, so that the set of all prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600121.png" /> can be partitioned into a finite number of classes and one can collect the prime ideals whose Kummer decomposition corresponds to one solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600122.png" /> of (2) into one class. The number of prime ideals with the property that some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600123.png" />, is finite, and they all are divisors of the discriminant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600124.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600125.png" />. Only infinite classes are of interest, so that classes for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600126.png" /> can be ignored.

In order to simplify the exposition, the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600127.png" /> is considered to be normal from now on. In such fields the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600128.png" /></td> </tr></table>

holds. Therefore, the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600129.png" /> not dividing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600130.png" /> is partitioned into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600131.png" /> classes, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600132.png" /> denotes the number of divisors. The class with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600133.png" /> is of special interest, in it

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600134.png" /></td> </tr></table>

A prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600135.png" /> in it has maximal number of prime divisors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600136.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600137.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

Such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600138.png" /> are said to split completely (or to be totally split), and their class is called the principal class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600139.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600140.png" />. It is a principal object of study in class field theory. In order to be able to define the principal class via (3) it is necessary to give a proof of the facts that prime ideals satisfying (1) do exist in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600141.png" /> and that there are infinitely many such ideals. Therefore a basic problem in class field theory is to define the principal class in terms of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600142.png" /> itself in such a way that its infinite nature would follow. This problem has been completely solved for Abelian extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600143.png" />.

In order to be able to understand the ideas of class field theory better, it is necessary to introduce the general concept of an ideal class group. Kummer's definition given above then corresponds to the modern concept of an absolute ideal class group. The general concept of an ideal class group which is used nowadays is due to H. Weber and T. Takagi, [[#References|[5]]].

Weber introduced the concept of the conductor of a class group. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600144.png" /> be an integral ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600145.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600146.png" /> be the subgroup of principal ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600147.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600148.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600149.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600150.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600151.png" /> be the subgroup of all ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600152.png" /> that are relatively prime with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600153.png" />. A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600154.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600155.png" /> can be regarded as a group of principal ideals, and a (generalized) class group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600156.png" /> is constructed in this way. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600157.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600158.png" /> one obtains Kummer's definition. In the general case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600159.png" /> consists of progressions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600160.png" /> whose residue classes form a subgroup of the whole multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600161.png" />. The order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600162.png" /> of this generalized class group satisfies the estimates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600163.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600164.png" /> is the order of the absolute class group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600165.png" /> is Euler's phi-function. For different conductors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600166.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600167.png" /> the class groups may be equivalent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600168.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600169.png" /> consists in fact of the same progressions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600170.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600171.png" />. If one agrees not to distinguish between equivalent class groups, then one obtains the concept of a class group according to Weber with a conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600172.png" /> that is the greatest common divisor of all equivalent conductors. A class field according to Weber is a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600173.png" /> in which only the prime ideals in its principal class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600174.png" />, and only these, split completely. Dirichlet's theorem on prime ideals in progressions, which is valid in every field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600175.png" />, implies that there are infinitely many prime ideals. Weber has shown that in certain particular cases the Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600176.png" /> of a class field and the class group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600177.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600178.png" /> are isomorphic.

A new point of view on class field theory, which is still valid, originated with D. Hilbert. He understood that between all relatively Abelian extensions of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600179.png" /> and all class fields for this field there must exist a one-to-one correspondence. This correspondence can be stated as follows. If for some conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600180.png" /> one constructs the Weber class group, then there is only one normal extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600181.png" /> in which the prime ideals of Weber's principal class, and only these, will split completely. Moreover, the Galois group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600182.png" /> is isomorphic to the Weber class group and the discriminant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600183.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600184.png" /> consists of the same prime ideals as the conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600185.png" />. The converse is also true: If an Abelian extension with Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600186.png" /> is given, then there exists a method (subsequently explicitly formulated by Takagi) by which one can uniquely construct a principal class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600187.png" /> such that the class group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600188.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600189.png" />, only the prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600190.png" /> split completely in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600191.png" />, and such that the conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600192.png" /> has the same prime divisors as the discriminant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600193.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600194.png" />. The  "duality"  was stated by Hilbert in 1900 as a hypothesis (he proved it only in special cases). In its general form it was proved by Takagi.

The next important stage of development of class group theory is related to the name of E. Artin, who revealed the special role of the canonical isomorphism between the Galois group and the ideal class group. He proved that this isomorphism is given by the [[Frobenius automorphism|Frobenius automorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600195.png" /> of an Abelian extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600196.png" />, defined as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600197.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600198.png" /> is the absolute norm of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600199.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600200.png" /> runs through all numbers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600202.png" /> is a prime ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600203.png" />. The automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600204.png" /> (nowadays denoted by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600205.png" /></td> </tr></table>

depends (in an appropriate group) only on the ideal class to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600206.png" /> belongs. It is multiplicative:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600207.png" /></td> </tr></table>

where the symbol at the right-hand side is understood as an automorphism of the class to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600208.png" /> belongs. With this in mind, Artin introduced the symbol

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600209.png" /></td> </tr></table>

on the entire group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600210.png" /> of ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600211.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600212.png" /> that are relatively prime with the conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600213.png" />. It is called the Artin symbol and realizes an isomorphism between the class group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600214.png" /> and the Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600215.png" />, whose explicit from is expressed in Artin's reciprocity law:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600216.png" /></td> </tr></table>

if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600217.png" /> (reciprocity as a correspondence between the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600218.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600219.png" />). From this one can obtain the classical form of the reciprocity law in the language of Kummer's power reciprocity symbol (one has to consider the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600220.png" />, cf. [[Reciprocity laws|Reciprocity laws]]). This form, in turn, implies the reciprocity law for Hilbert's symbol. In all three forms the reciprocity law is regarded as an isomorphism of groups, and the symbols of Artin, Kummer and Hilbert are regarded as group elements realizing this isomorphism. However, each of them also has a numerical value, which is equal to some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600221.png" />-th root of unity. Therefore one can formulate the reciprocity law as follows. Given the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600222.png" />, it is required to find the value of the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600223.png" /> reciprocal to it, i.e. to exhibit the explicit form of the function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600224.png" /> defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600225.png" /></td> </tr></table>

In this form the law first appeared in the work of C.F. Gauss (cf. [[Gauss reciprocity law|Gauss reciprocity law]]) for quadratic fields and in the work of Kummer for cyclotomic fields of prime degree. The study of the reciprocity law in this form was subsequently continued by C.G.J. Jacobi, F.G.M. Eisenstein, Hilbert, H. Hasse, and others, but they only obtained partial results. I.R. Shafarevich [[#References|[6]]] solved this problem in its general form in 1948, basing himself on the idea of establishing a connection between the arithmetic definition of the symbol

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600226.png" /></td> </tr></table>

and the analytic definition of an Abelian differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600227.png" /> on a Riemann surface. He gave a construction of

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600228.png" /></td> </tr></table>

which precisely corresponds to the determination of the residue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600229.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600230.png" />-adic field. For this purpose he introduced <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600231.png" />-functions, also called Shafarevich functions, in terms of which he also obtained an explicit form of the reciprocity law in the general case.

In the late 1920s Hasse introduced the idea of doing class field theory for one prime ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600232.png" /> at a time, and reformulated and proved many theorems for an Abelian extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600233.png" /> of a local field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600234.png" />. At first the idea was of secondary importance only (the results were obtained as a consequence of the local theory), but at the end of the 1930s C. Chevalley proved the extraordinary importance of the local point of view in the creation of class field theory. He introduced the concept of an idèle group and formulated general class field theory from the local point of view. Following this the local-global principle became established in class field theory. Subsequently it was extended and refined (cf. [[#References|[5]]]), and as a result Abelian class field theory took a structured and completed form. The problem of creating non-Abelian class field theory for normal extensions with non-Abelian Galois group remains.

The exposition above relates mainly to the qualitative aspects of algebraic number theory. In questions of quantitative estimation and methods algebraic number theory is intimately connected with analytic number theory. It is also based, to a large extent, on properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600235.png" />-functions and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600236.png" />-functions of algebraic fields.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Z.I. Borevich,   I.R. Shafarevich,   "Number theory" , Acad. Press  (1966)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.M. Postnikov,   "Fundamentals of Galois theory" , Noordhoff  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Weyl,   "Algebraic theory of numbers" , Princeton Univ. Press  (1959)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Lang,   "Algebraic number theory" , Addison-Wesley  (1970)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1967)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  I.R. Shafarevich,   "General reciprocity laws"  ''Mat. Sb.'' , '''26 (68)''' :  1  (1950)  pp. 113–146  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  "Hilbert's problems"  ''Bull. Amer. Math. Soc.'' , '''8'''  (1902)  pp. 437–479  (Translated from German)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  E. Artin,   J. Tate,   "Class field theory" , Benjamin  (1967)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A. Weil,   "Basic number theory" , Springer  (1974)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  S. Lang,   "Algebraic numbers" , Addison-Wesley  (1964)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  N. Bourbaki,   "Commutative algebra" , Addison-Wesley  (1966)  (Translated from French)</TD></TR></table>



====Comments====
An algebraic number field of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600237.png" /> is a degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600238.png" /> extension of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600239.png" /> of rational numbers. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600240.png" /> is an algebraic integer if it is integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600241.png" />, i.e. if it is a root of a monic polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600242.png" />. These form a free Abelian subgroup of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600243.png" />, called the ring of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600244.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600245.png" /> and also the principal order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600246.png" />; a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600247.png" /> is an integral basis, also called a minimal basis.

For Kummer's approach to the Fermat problem cf. [[#References|[a3]]]. The link (equivalence) between Kummer's ideal numbers and ideals comes from the fact that for the prime ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600248.png" /> there is a field extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600249.png" /> in which they become principal ideals.

The relative degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600250.png" /> of an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600251.png" /> is the degree of the residue field extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600252.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600253.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600254.png" />.

For more details, definitions, newer results, and more precise statements of much of the above cf. various more specified articles in this Encyclopaedia. In particular the articles [[Algebraic number|Algebraic number]]; [[Unit|Unit]]; [[Extension of a field|Extension of a field]]; [[Regulator of an algebraic number field|Regulator of an algebraic number field]]; [[Discriminant|Discriminant]]; [[Frobenius automorphism|Frobenius automorphism]] (also for Artin symbol), [[Kummer theorem|Kummer theorem]] on decomposition of ideals, and especially [[Class field theory|class field theory]].

The ideal class groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600255.png" /> are also often known as ray class groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600256.png" />, and the corresponding fields as ray class fields. The class field corresponding to the ray class group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600257.png" /> is the Hilbert class field. The definitions above ignore some problems with the Archimedean primes; in particular the principal ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600258.png" /> making up <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600259.png" /> must have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600260.png" /> totally positive, i.e. such that all real conjugates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600261.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011600/a011600262.png" />.

The discussion of class field theory above (up to the mention of idèles) is in terms of the older ideal-theoretic formulation as in e.g. Hasse's Zahlbericht [[#References|[a1]]], which in fact can be advantageously consulted for more details and precise statements. For a discussion of the relations between the ideal-theoretic formulations and the modern idèle based class field theory [[#References|[a2]]] is recommended.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Hasse,   "Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraïsche Zahlkörper" , '''I.II''' , Physika Verlag  (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Neukirch,   "Class field theory" , Springer  (1986)  pp. Chapt. 4, Sect. 8</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Pollard,   "The theory of algebraic numbers" , Math. Assoc. Amer.  (1950)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Weiss,   "Algebraic number theory" , McGraw-Hill  (1963)  pp. Sects. 4–9</TD></TR></table>