Discriminant

The discriminant of a polynomial , , whose roots are is the product

The discriminant vanishes if and only if the polynomial has multiple roots. The discriminant is symmetric with respect to the roots of the polynomial and may therefore be expressed in terms of the coefficients of this polynomial.

The discriminant of a quadratic polynomial is ; the discriminant of the polynomial (the roots of which can be computed by the Cardano formula) is . If is a polynomial over a field of characteristic zero, then

where is the resultant of and its derivative . The derivative of a polynomial with coefficients from an arbitrary field is the polynomial .

References

[1]	A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian) MR0945393 MR0926059 MR0778202 MR0759341 MR0628003 MR0384363 Zbl 0237.13001

References

[a1]	S. Lang, "Algebra" , Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[a2]	B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German) MR0263582 Zbl 1032.00001 Zbl 1032.00002

The discriminant of a form sesquilinear with respect to an automorphism in a basis is the element of the ring equal to

(*)

where is a fixed basis of a free -module of finite rank over the commutative ring (with a unit element). If is another basis in and if

is the transition matrix from to , then

If has no zero divisors, then for not to be degenerate it is necessary and sufficient that

If are elements arbitrarily chosen from , then the element of defined by (*) is called the discriminant of with respect to the system . Let have no zero divisors and let be a non-degenerate sesquilinear form. Then, for a system of elements from to be free it is necessary and sufficient that . Here, form a basis in if and only if and are associated in for some basis in .

References

[1]	N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) MR2333539 MR2327161 MR2325344 MR2284892 MR2272929 MR0928386 MR0896478 MR0782297 MR0782296 MR0722608 MR0682756 MR0643362 MR0647314 MR0610795 MR0583191 MR0354207 MR0360549 MR0237342 MR0205211 MR0205210
[2]	J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) Zbl 0221.20056

V.L. Popov

The discriminant of a system of elements of a field is one of the most important constructions in the theory of field extensions. Let be a finite extension of a field of degree . The mapping from into :

where and is the trace of an element , is a symmetric bilinear form on the field , which is regarded as a linear space over . The discriminant of this bilinear form (cf. Discriminant of a form) with respect to a system of elements from is said to be the discriminant of the system and is denoted by . In particular, if the system is a basis of over , its discriminant is called the discriminant of the basis of over . The discriminants of two bases differ by a factor which is the square of some non-zero element of . The discriminant of any basis of over is non-zero if and only if the extension is separable (cf. Separable extension). If is a polynomial of degree which is the minimal polynomial of the element from the separable extension , then coincides with the discriminant of the polynomial . The definitions above can also be applied to an arbitrary finite-dimensional associative algebra over a field (see 4) below).

In the case of a separable extension the discriminant of the basis may be calculated by the formula

where are all different imbeddings of in a given algebraic closure of which leave fixed.

Let be the field of rational numbers, let be an algebraic number field and let be some lattice of rank in . Then, for any two bases of the values of discriminant are identical, and this common value is known as the discriminant of the lattice . If coincides with the ring of integers of the field , the discriminant of is simply called the discriminant of the field and is denoted by ; this quantity is an important characteristic of . For instance, if permits real and complex imbeddings in the field of complex numbers, then

where is Dedekind's zeta-function; is the number of divisor classes, is the regulator of (cf. Regulator of an algebraic number field) and is the number of roots of unity in . By virtue of the estimate

. For a quadratic field , where is a square-free rational integer, , one has the formulas

For a cyclotomic field , where is a primitive -th root of unity, one has

the minus sign being taken if or (), while the plus sign is taken in the other cases.

This definition of the discriminant of a lattice in an algebraic number field may be generalized to the case when is the field of fractions of a Dedekind ring , and is a finite separable extension of of degree . Let be the integral closure of the ring in and let be an arbitrary fractional ideal in . Then the -module generated by all discriminants of the form , where run through all possible bases of over and lying in , is called the discriminant of the ideal . will then be a fractional ideal of , and the equality , where is the norm of the ideal , is valid. The discriminant is identical with the norm of the different of the ring over .

References

[1]	Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) MR0195803 Zbl 0145.04902
[2]	S. Lang, "Algebraic numbers" , Addison-Wesley (1964) MR0160763 Zbl 0211.38501
[3]	O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975) MR0389876 MR0384768 Zbl 0313.13001
[4]	N. Jacobson, "The theory of rings" , Amer. Math. Soc. (1943) MR0008601 Zbl 0060.07302

V.L. Popov

The discriminant of an algebra is the discriminant of the symmetric bilinear form , where are elements of the finite-dimensional associative algebra over a field , while is the principal trace of the element , which is defined as follows: Let be some basis of the algebra , let be a purely transcendental extension of the field formed with algebraically independent elements , and let be the corresponding scalar extension of the algebra . An element is then said to be a generic element of the algebra , while the minimal polynomial (over ) of the element is known as the minimal polynomial of the algebra . Let

be the minimal polynomial of the algebra ; the coefficients are in fact polynomials from . If () is an arbitrary element of , then is said to be the principal trace of the element , is said to be its principal norm, while the polynomial is known as its principal polynomial. For a given element the coefficients of the principal polynomial are independent of the basis chosen; for this reason the bilinear form on which was mentioned above is defined invariantly, while its discriminant is defined up to a multiplicative factor which is the square of a non-zero element of . The algebra is separable (cf. Separable algebra) if and only if its discriminant is non-zero.

References

[1]	N. Jacobson, "The theory of rings" , Amer. Math. Soc. (1943) MR0008601 Zbl 0060.07302

E.N. Kuz'min

Comments

Let be a global field (an algebraic number field or a function field in one variable) or a local field, and let be a finite separable field extension. Let and be the rings of integers (principal orders) of and , respectively. Let where is the trace function.

(Let be a finite-dimensional commutative algebra over a field and an element of . Choose a basis of over . Then multiplication with , , is given by a certain matrix . One now defines, the trace, norm and characteristic polynomial of as the trace, determinant and characteristic polynomial of the matrix :

The set is a fractional ideal of . Its inverse in the group of fractional ideals of the Dedekind ring is called the different of the field extension , and is denoted by . Sometimes (if ) it is called the relative different, and the (absolute) different of is then . If is a tower of field extensions one has the chain theorem for differents, also called multiplicativity of differents in a tower:

The ideal is an integral ideal of (i.e. ) and it is related to the discriminant of the field extension by

For the different to be divisible by a prime ideal of it is necessary and sufficient that with , where . This is Dedekind's discriminant theorem. Hence a prime ideal of is ramified in if and only if divides the discriminant of .

Given an additive subgroup of , its complementary set (relative to the trace) is defined by

It is also an additive subgroup of . Thus, the different of is the inverse of the complementary set of the ring of integers of .

More generally one defines the different of an ideal in as the inverse of its complementary set: . It is again a (fractional) ideal of . The different of an element in is defined as where is the derivative of the characteristic polynomial of the element in . If , then the different is in and is an integral basis of over if and only if .

Let now be a finite extension of global fields. For each prime ideal of let be the corresponding local field (the completion of with respect to the -adic topology on ). As before, if is a prime ideal of , is the prime ideal of underneath it: . Then one has for the local and global differents that

where an ideal of is identified with its completion in , and where the statement is supposed to include that all but finitely many of the factors on the right-hand side are 1, i.e. unit ideals ( for almost-all ).

Let now be a Dedekind integral domain with quotient field and let be a central simple algebra over (i.e. is a finite-dimensional associative algebra over with no ideals except and and the centre of is ). Then there is a separable normal extension such that (as -algebras), where is the algebra of ()-matrices over . (Such an is called a splitting field for .) For each consider the element . The trace of this matrix is an element of (not just of ); it is called the reduced trace and is denoted by . (Its definition is also independent of the choice of and .) Similarly one defines the reduced norm, , as .

An -lattice in is an -submodule of that is finitely generated over and is such that . An -lattice that is a subring and contains is called an order. A maximal order is an order that is not contained in any other. These always exist but may be non-unique. (These three definitions hold for any separable associative algebra over not just for central simple ones.)

Let be a maximal order in . The different of in this setting is defined by . The discriminant of a central simple algebra is the ideal . It does not depend on the choice of the maximal order .

References

[a1]	N. Jacobson, "Lectures in abstract algebra" , 3. Theory of fields and galois theory , v. Nostrand (1964) MR0172871 Zbl 0124.27002
[a2]	S. Lang, "Algebraic numbers" , Addison-Wesley (1964) MR0160763 Zbl 0211.38501
[a3]	A. Weil, "Basic number theory" , Springer (1967) MR0234930 Zbl 0176.33601
[a4]	E. Weiss, "Algebraic number theory" , McGraw-Hill (1963) pp. Sects. 4–9 MR0159805 Zbl 0115.03601