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− | A commutative, associative ring containing a unit in which the set of non-zero elements is not empty and forms a group under multiplication (cf. [[Associative rings and algebras|Associative rings and algebras]]). A field may also be characterized as a simple non-zero commutative, associative ring containing a unit. Examples of fields: the field of rational numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f0400901.png" />, the field of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f0400902.png" />, the field of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f0400903.png" />, finite fields (see [[Galois field|Galois field]]), and the field of fractions of an integral domain.
| + | {{MSC|12}} |
| + | {{TEX|done}} |
| | | |
− | A subfield of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f0400904.png" /> is a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f0400905.png" /> which itself is a field under the operations of addition and multiplication defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f0400906.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f0400907.png" /> is some automorphism of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f0400908.png" />, then the set | + | A ''field'' is a commutative, associative ring containing a unit in which the set of |
| + | non-zero elements is not empty and forms a group under multiplication |
| + | (cf. |
| + | [[Associative rings and algebras|Associative rings and algebras]]). A |
| + | field may also be characterized as a simple non-zero commutative, |
| + | associative ring containing a unit. Examples of fields: the field of |
| + | rational numbers $\Q$, the field of real numbers $\R$, the field of |
| + | complex numbers $\C$, finite fields (see |
| + | [[Galois field|Galois field]]), and the field of fractions of an |
| + | integral domain. |
| | | |
− | <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/f/f040/f040090/f0400909.png" /></td> </tr></table>
| + | A subfield of a field $K$ is a subset $M\subset K$ which itself is a field |
| + | under the operations of addition and multiplication defined in |
| + | $K$. For example, if $\def\s{\sigma} \s$ is some automorphism of a field $K$, then the |
| + | set |
| + | $$K^\s = \{ x\in K:\s(x)=x \}$$ |
| + | is a subfield in $K$. If $M$ and $N$ are subfields of a |
| + | field $K$, then their intersection $M\cap N$ is a subfield in $K$; also, |
| + | there exists a smallest subfield $MN$ in the field $K$ that contains |
| + | $M$ and $N$, called the composite of the fields $M$ and $N$ (in |
| + | $K$). Each field contains a unique prime subfield (i.e. one not |
| + | containing proper subfields). |
| | | |
− | is a subfield in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009012.png" /> are subfields of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009013.png" />, then their intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009014.png" /> is a subfield in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009015.png" />; also, there exists a smallest subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009016.png" /> in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009017.png" /> that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009019.png" />, called the composite of the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009021.png" /> (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009022.png" />). Each field contains a unique prime subfield (i.e. one not containing proper subfields). | + | Any field homomorphism is an imbedding. For an arbitrary field $K$ |
| + | there exists a unique homomorphism $\def\phi{\varphi} \phi: \Z \to K$ which maps the unit of the ring |
| + | $\Z$ to the unit of the field $K$. If $\ker \phi = 0$, $K$ is called a field of |
| + | characteristic zero. In that case, the prime subfield of $K$ coincides |
| + | with the field of fractions of the ring $\phi(\Z)$ and is isomorphic to the |
| + | field $\Q$. If $\ker \phi \ne 0$, then $\ker\phi = p\Z$ for a certain prime $p$. This $p$ is |
| + | called the characteristic of the field $K$. The prime subfield of $K$ |
| + | coincides in that case with $\phi(\Z) \cong \Z/p\Z$. |
| | | |
− | Any field homomorphism is an imbedding. For an arbitrary field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009023.png" /> there exists a unique homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009024.png" /> which maps the unit of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009025.png" /> to the unit of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009028.png" /> is called a field of characteristic zero. In that case, the prime subfield of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009029.png" /> coincides with the field of fractions of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009030.png" /> and is isomorphic to the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009031.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009032.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009033.png" /> for a certain prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009034.png" />. This <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009035.png" /> is called the characteristic of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009036.png" />. The prime subfield of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009037.png" /> coincides in that case with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009038.png" />.
| + | If $k$ is a subfield of a field $K$, $K$ is called an extension of the |
| + | field $k$. Let $Y$ be some subset in $K$. Then the field $k(Y)$ is |
| + | defined as the smallest subfield of $K$ that contains $Y$ and $k$. It |
| + | is said that $k(Y)$ is obtained from $k$ by adjoining the elements from |
| + | the set $Y$. |
| | | |
− | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009039.png" /> is a subfield of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009041.png" /> is called an extension of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009042.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009043.png" /> be some subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009044.png" />. Then the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009045.png" /> is defined as the smallest subfield of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009046.png" /> that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009048.png" />. It is said that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009049.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009050.png" /> by adjoining the elements from the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009051.png" />.
| + | Basic problems in the theory of fields consist of giving a description |
| + | of all subfields of a given field, of all fields containing a given |
| + | field, i.e. overfields (see |
| + | [[Extension of a field|Extension of a field]]), to examine all |
| + | imbeddings of a field in some other field, to classify fields up to an |
| + | isomorphism, and to examine the automorphism group of a given field. |
| | | |
− | Basic problems in the theory of fields consist of giving a description of all subfields of a given field, of all fields containing a given field, i.e. overfields (see [[Extension of a field|Extension of a field]]), to examine all imbeddings of a field in some other field, to classify fields up to an isomorphism, and to examine the automorphism group of a given field.
| + | A field $K$ is said to be finitely generated over a subfield $k$ if |
| + | there exists a finite set $Y\subset K$ such that $K=k(Y)$. Any such field can be |
| + | interpreted as the field of rational functions, $k(X)$, of a certain |
| + | irreducible algebraic variety $X$ defined over $k$. |
| + | [[Algebraic geometry|Algebraic geometry]] deals, among other things, |
| + | with the study of such fields. In particular, the classification of |
| + | such fields is equivalent to the birational classification of |
| + | irreducible algebraic varieties, and the problem of finding the group |
| + | of automorphisms of a field $K=k(X)$ that leave all elements of the field |
| + | $k$ invariant is equivalent to finding all birational automorphisms of |
| + | the variety $X$ defined over $k$. |
| + | [[Galois theory|Galois theory]] deals with finite separable extensions |
| + | (cf. |
| + | [[Separable extension|Separable extension]]) of arbitrary fields. An |
| + | important part in number theory is played by the finite extensions of |
| + | the field $\Q$, which are called algebraic number fields. |
| + | [[Algebraic number theory|Algebraic number theory]] deals with these |
| + | fields. |
| | | |
− | A field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009052.png" /> is said to be finitely generated over a subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009053.png" /> if there exists a finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009054.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009055.png" />. Any such field can be interpreted as the field of rational functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009056.png" />, of a certain irreducible algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009057.png" /> defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009058.png" />. [[Algebraic geometry|Algebraic geometry]] deals, among other things, with the study of such fields. In particular, the classification of such fields is equivalent to the birational classification of irreducible algebraic varieties, and the problem of finding the group of automorphisms of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009059.png" /> that leave all elements of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009060.png" /> invariant is equivalent to finding all birational automorphisms of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009061.png" /> defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009062.png" />.
| + | Field theory also deals with fields having certain additional |
| + | structures, such as differential fields, topological fields, ordered |
| + | fields, formally real and formally $p$-adic fields, etc. |
| | | |
− | [[Galois theory|Galois theory]] deals with finite separable extensions (cf. [[Separable extension|Separable extension]]) of arbitrary fields. An important part in number theory is played by the finite extensions of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009063.png" />, which are called algebraic number fields. [[Algebraic number theory|Algebraic number theory]] deals with these fields.
| + | Field theory originated (within the framework of the theory of |
| + | algebraic equations) in the middle of the 19th century. Papers by |
| + | E. Galois and J.L. Lagrange on group theory and by C.F. Gauss on |
| + | number theory made it clear that one had to examine the nature of |
| + | number systems themselves. The concept of a field was put forward in |
| + | papers by L. Kronecker and R. Dedekind. Dedekind introduced the |
| + | concept of a field, which he originally called a "rational |
| + | domainrational domain" . Dedekind's theory was published in the |
| + | comments and supplements to P.G. Lejeune-Dirichlet's Zahlentheorie. In |
| + | them, Dedekind substantially supplemented and extended the theory of |
| + | numbers, the theory of ideals and the theory of finite fields. The |
| + | term "field" first appeared in the edition of this book in 1871. |
| | | |
− | Field theory also deals with fields having certain additional structures, such as differential fields, topological fields, ordered fields, formally real and formally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040090/f04009064.png" />-adic fields, etc.
| + | ====Comments==== |
− | | + | The German term for "field" is "Körper" and this is |
− | Field theory originated (within the framework of the theory of algebraic equations) in the middle of the 19th century. Papers by E. Galois and J.L. Lagrange on group theory and by C.F. Gauss on number theory made it clear that one had to examine the nature of number systems themselves. The concept of a field was put forward in papers by L. Kronecker and R. Dedekind. Dedekind introduced the concept of a field, which he originally called a "rational domainrational domain" . Dedekind's theory was published in the comments and supplements to P.G. Lejeune-Dirichlet's Zahlentheorie. In them, Dedekind substantially supplemented and extended the theory of numbers, the theory of ideals and the theory of finite fields. The term "field" first appeared in the edition of this book in 1871.
| + | of course the term used in |
− | | + | {{Cite|Le}}. The edition cited here as |
− | ====References====
| + | {{Cite|Le}} is a corrected reprint of the 4th edition |
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Eléments de mathematique. Algèbre" , Masson (1981) pp. Chapt. 4–7</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.L. van der Waerden, "Algebra" , '''1–2''' , Springer (1967–1971) (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''1''' , Springer (1975)</TD></TR></table>
| + | (Braunschweig, 1893); the 1871 edition was the second. |
| | | |
− |
| |
− |
| |
− | ====Comments====
| |
− | The German term for "field" is "Körper" and this is of course the term used in [[#References|[a2]]]. The edition cited here as [[#References|[a2]]] is a corrected reprint of the 4th edition (Braunschweig, 1893); the 1871 edition was the second.
| |
| | | |
| ====References==== | | ====References==== |
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Jacobson, "Lectures in abstract algebra" , '''1. Basic concepts''' , Springer (1975)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.G. Lejeune-Dirichlet, "Zahlentheorie" , Chelsea, reprint (1968)</TD></TR></table>
| + | {| |
| + | |- |
| + | |valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Eléments de mathematique. Algèbre", Masson (1981) pp. Chapt. 4–7 {{MR|0643362}} {{ZBL|0498.12001}} |
| + | |- |
| + | |valign="top"|{{Ref|Ja}}||valign="top"| N. Jacobson, "Lectures in abstract algebra", '''1. Basic concepts''', Springer (1975) {{MR|0041102}} {{ZBL|0326.00001}} |
| + | |- |
| + | |valign="top"|{{Ref|La}}||valign="top"| S. Lang, "Algebra", Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}} |
| + | |- |
| + | |valign="top"|{{Ref|Le}}||valign="top"| P.G. Lejeune-Dirichlet, "Zahlentheorie", Chelsea, reprint (1968) {{MR|}} {{ZBL|25.0252.01}} {{ZBL|03.0063.01}} |
| + | |- |
| + | |valign="top"|{{Ref|Wa}}||valign="top"| B.L. van der Waerden, "Algebra", '''1–2''', Springer (1967–1971) (Translated from German) {{MR|0263582}} {{ZBL|1032.00001}} {{ZBL|1032.00002}} |
| + | |- |
| + | |valign="top"|{{Ref|ZaSa}}||valign="top"| O. Zariski, P. Samuel, "Commutative algebra", '''1''', Springer (1975) {{MR|0389876}} {{MR|0384768}} {{ZBL|0313.13001}} |
| + | |- |
| + | |} |
2020 Mathematics Subject Classification: Primary: 12-XX [MSN][ZBL]
A field is a commutative, associative ring containing a unit in which the set of
non-zero elements is not empty and forms a group under multiplication
(cf.
Associative rings and algebras). A
field may also be characterized as a simple non-zero commutative,
associative ring containing a unit. Examples of fields: the field of
rational numbers $\Q$, the field of real numbers $\R$, the field of
complex numbers $\C$, finite fields (see
Galois field), and the field of fractions of an
integral domain.
A subfield of a field $K$ is a subset $M\subset K$ which itself is a field
under the operations of addition and multiplication defined in
$K$. For example, if $\def\s{\sigma} \s$ is some automorphism of a field $K$, then the
set
$$K^\s = \{ x\in K:\s(x)=x \}$$
is a subfield in $K$. If $M$ and $N$ are subfields of a
field $K$, then their intersection $M\cap N$ is a subfield in $K$; also,
there exists a smallest subfield $MN$ in the field $K$ that contains
$M$ and $N$, called the composite of the fields $M$ and $N$ (in
$K$). Each field contains a unique prime subfield (i.e. one not
containing proper subfields).
Any field homomorphism is an imbedding. For an arbitrary field $K$
there exists a unique homomorphism $\def\phi{\varphi} \phi: \Z \to K$ which maps the unit of the ring
$\Z$ to the unit of the field $K$. If $\ker \phi = 0$, $K$ is called a field of
characteristic zero. In that case, the prime subfield of $K$ coincides
with the field of fractions of the ring $\phi(\Z)$ and is isomorphic to the
field $\Q$. If $\ker \phi \ne 0$, then $\ker\phi = p\Z$ for a certain prime $p$. This $p$ is
called the characteristic of the field $K$. The prime subfield of $K$
coincides in that case with $\phi(\Z) \cong \Z/p\Z$.
If $k$ is a subfield of a field $K$, $K$ is called an extension of the
field $k$. Let $Y$ be some subset in $K$. Then the field $k(Y)$ is
defined as the smallest subfield of $K$ that contains $Y$ and $k$. It
is said that $k(Y)$ is obtained from $k$ by adjoining the elements from
the set $Y$.
Basic problems in the theory of fields consist of giving a description
of all subfields of a given field, of all fields containing a given
field, i.e. overfields (see
Extension of a field), to examine all
imbeddings of a field in some other field, to classify fields up to an
isomorphism, and to examine the automorphism group of a given field.
A field $K$ is said to be finitely generated over a subfield $k$ if
there exists a finite set $Y\subset K$ such that $K=k(Y)$. Any such field can be
interpreted as the field of rational functions, $k(X)$, of a certain
irreducible algebraic variety $X$ defined over $k$.
Algebraic geometry deals, among other things,
with the study of such fields. In particular, the classification of
such fields is equivalent to the birational classification of
irreducible algebraic varieties, and the problem of finding the group
of automorphisms of a field $K=k(X)$ that leave all elements of the field
$k$ invariant is equivalent to finding all birational automorphisms of
the variety $X$ defined over $k$.
Galois theory deals with finite separable extensions
(cf.
Separable extension) of arbitrary fields. An
important part in number theory is played by the finite extensions of
the field $\Q$, which are called algebraic number fields.
Algebraic number theory deals with these
fields.
Field theory also deals with fields having certain additional
structures, such as differential fields, topological fields, ordered
fields, formally real and formally $p$-adic fields, etc.
Field theory originated (within the framework of the theory of
algebraic equations) in the middle of the 19th century. Papers by
E. Galois and J.L. Lagrange on group theory and by C.F. Gauss on
number theory made it clear that one had to examine the nature of
number systems themselves. The concept of a field was put forward in
papers by L. Kronecker and R. Dedekind. Dedekind introduced the
concept of a field, which he originally called a "rational
domainrational domain" . Dedekind's theory was published in the
comments and supplements to P.G. Lejeune-Dirichlet's Zahlentheorie. In
them, Dedekind substantially supplemented and extended the theory of
numbers, the theory of ideals and the theory of finite fields. The
term "field" first appeared in the edition of this book in 1871.
The German term for "field" is "Körper" and this is
of course the term used in
[Le]. The edition cited here as
[Le] is a corrected reprint of the 4th edition
(Braunschweig, 1893); the 1871 edition was the second.
References
[Bo] |
N. Bourbaki, "Eléments de mathematique. Algèbre", Masson (1981) pp. Chapt. 4–7 MR0643362 Zbl 0498.12001
|
[Ja] |
N. Jacobson, "Lectures in abstract algebra", 1. Basic concepts, Springer (1975) MR0041102 Zbl 0326.00001
|
[La] |
S. Lang, "Algebra", Addison-Wesley (1974) MR0783636 Zbl 0712.00001
|
[Le] |
P.G. Lejeune-Dirichlet, "Zahlentheorie", Chelsea, reprint (1968) Zbl 25.0252.01 Zbl 03.0063.01
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[Wa] |
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[ZaSa] |
O. Zariski, P. Samuel, "Commutative algebra", 1, Springer (1975) MR0389876 MR0384768 Zbl 0313.13001
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