Discriminant

2020 Mathematics Subject Classification: Primary: 08-XX Secondary: 11-XX12-XX13-XX16-XX [MSN][ZBL]

The notion of discriminant can have different meanings, depending on the context.

Discriminant of a polynomial

Cf. [Ku], [La], [Wa].

The discriminant of a polynomial $f(x)=a_0 x^n+a_1 x^{n-1}+\cdots + a_n$, $a_0 \ne 0$, whose roots are $\def\a{ {\alpha}}\a_1,\dots,\a_n$ is the product

$$D(f) = a_0^{2n-2} \prod_{n\ge i > j \ge 1}(\a_i - \a_j)^2.$$ 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 $ax^2+bx+c$ is $b^2-4ac$; the discriminant of the polynomial $x^3+px+q$ (the roots of which can be computed by the Cardano formula) is $-27 q^2-4p^3$. If $f(x)$ is a polynomial over a field of characteristic zero, then

$$D(f) = (-1)^{n(n-1)/2}a_0^{-1} R(f,f')$$ where $R(f,f')$ is the resultant of $f(x)$ and its derivative $f'(x)$. The derivative of a polynomial $f(x) = a_0 x^n + a_1 x^{n-1}+ \cdots + a_n$ with coefficients from an arbitrary field is the polynomial $n a_0 x^{n-1} + (n-1)a_1 x^{n-2}+ \cdots + a_{n-1}$.

Discriminant of a sesquilinear form

Cf. [Bo], [Di]

The discriminant of a form $f$ sesquilinear with respect to an automorphism $\def\s{ {\sigma}}$ in a basis $(v) = \{v_1,\dots,v_n\}$ is the element of the ring $A$ equal to

$$D_f(v_1,\dots,v_n) = {\rm det}\begin{pmatrix} f(v_1,v_1) & \cdots &f(v_1,v_n)\\ \vdots &\ddots &\vdots \\ f(v_n,v_1)&\cdots&f(v_n,v_n)\end{pmatrix},\label{*}$$ where $(v)$ is a fixed basis of a free $A$-module $E$ of finite rank over the commutative ring $A$ (with a unit element). If $(w) = \{w_1,dots,w_n\}$ is another basis in $E$ and if

$$C=\begin{pmatrix} c_{1,1} & \cdots &c_{1,n}\\ \vdots &\ddots &\vdots \\ c_{n,1}&\vdots&c_{n,n}\end{pmatrix}$$ is the transition matrix from $(v)$ to $(w)$, then

$$D_f(w_1,\dots,w_n) = (\det C)(\det C)^\s D_f(v_1,\dots,v_n).$$ If $A$ has no zero divisors, then for $f$ not to be degenerate it is necessary and sufficient that

$$D_f(v_1,\dots,v_n) \ne 0.$$ If $v_1,\dots,v_n$ are $n$ elements arbitrarily chosen from $E$, then the element $D_f(v_1,\dots,v_n)$ of $A$ defined by (1) is called the discriminant of $f$ with respect to the system $v_1,\dots,v_n$. Let $A$ have no zero divisors and let $f$ be a non-degenerate sesquilinear form. Then, for a system of elements $v_1,\dots,v_n$ from $E$ to be free it is necessary and sufficient that $D_f(v_1,\dots,v_n) \ne 0$. Here, $v_1,\dots,v_n$ form a basis in $E$ if and only if $D_f(v_1,\dots,v_n) $ and $D_f(u_1,\dots,u_n) $ are associated in $A$ for some basis $u_1,\dots,u_n $ in $E$.

V.L. Popov

Discriminant of an algebra

Cf. [Ja].

The discriminant of an algebra $A$ is the discriminant of the symmetric bilinear form $\def\Tr{ {\rm Tr}}(x,y)=\Tr(xy)$, where $x,y$ are elements of the finite-dimensional associative algebra $A$ over a field $F$, while $\Tr(a)$ is the principal trace of the element $a\in A$, which is defined as follows: Let $e_1,\dots,e_n$ be some basis of the algebra $A$, let $\Phi=F(\xi_1,\dots,\xi_n)$ be a purely transcendental extension of the field $F$ formed with algebraically independent elements $\xi_1,\dots,\xi_n$, and let $\def\Ph{ {\Phi}}A_\Ph = A_F\otimes \Ph$ be the corresponding scalar extension of the algebra $A$. An element $x=\xi_1 e_1+\cdots+\xi_n e_n\in A_\Ph$ is then said to be a generic element of the algebra $A$, while the minimal polynomial (over $\Ph$) of the element $x$ is known as the minimal polynomial of the algebra $A$. Let

$$g(t,\xi) = t^r -m_1(\xi) t^{r-1} +\cdots + (-1)^r m_r(\xi)$$ be the minimal polynomial of the algebra $A$; the coefficients $m_i(\xi)$ are in fact polynomials from $F[\xi_1,\dots,\xi_n]$. If $\a = \a_1 e_1+\cdots+\a_n e_n$ ($\a_i \in F$) is an arbitrary element of $A$, then $m_1(\a_1,\dots,\a_n) = \Tr(\a)$ is said to be the principal trace of the element $\a$, $m_r(\a_1,\dots,\a_n) = {\rm N}(\a)$ is said to be its principal norm, while the polynomial $g(t,\a_1,\dots,\a_n)$ is known as its principal polynomial. For a given element $\a\in A$ the coefficients of the principal polynomial are independent of the basis chosen; for this reason the bilinear form $(x,y)$ on $A$ 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 $F$. The algebra $A$ is separable (cf. Separable algebra) if and only if its discriminant is non-zero.

E.N. Kuz'min

Discriminant of a field

Cf. [BoSh], [La2], [Ja], [ZaSe].

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

$$(x,y)\mapsto \def\Tr{ {\rm Tr}} \Tr(xy),$$ where $x,y\in K$ and $\Tr \a$ is the trace of an element $\a\in K$, is a symmetric bilinear form on the field $K$, which is regarded as a linear space over $k$. The discriminant of this bilinear form (cf. Discriminant of a form) with respect to a system of elements $w_1,\dots,w_m $ from $K$ is said to be the discriminant of the system $w_1,\dots,w_m $ and is denoted by $D(w_1,\dots,w_m) $. In particular, if the system is a basis of $K$ over $k$, its discriminant is called the discriminant of the basis of $K$ over $k$. The discriminants of two bases differ by a factor which is the square of some non-zero element of $k$. The discriminant of any basis of $K$ over $k$ is non-zero if and only if the extension $K/k$ is separable (cf. Separable extension). If $f_x(t)$ is a polynomial of degree $m$ which is the minimal polynomial of the element $x$ from the separable extension $K/k$, then $D(1,x,\dots,x^m)$ coincides with the discriminant of the polynomial $f_x(t)$. 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 $K/k$ the discriminant of the basis $w_1,\dots,w_n $ may be calculated by the formula

$$D(w_1,\dots,w_n) = (\det (\s_i(w_j)))^2, $$ where $\s_1,\dots,\s_n $ are all different imbeddings of $K$ in a given algebraic closure of $k$ which leave $k$ fixed.

Let $k=\Q$ be the field of rational numbers, let $K$ be an algebraic number field and let $M$ be some lattice of rank $n$ in $K$. Then, for any two bases of $M$ the values of discriminant are identical, and this common value is known as the discriminant of the lattice $M$. If $M$ coincides with the ring of integers of the field $K$, the discriminant of $M$ is simply called the discriminant of the field $K$ and is denoted by $D_K$; this quantity is an important characteristic of $K$. For instance, if $K$ permits $s$ real and $2t$ complex imbeddings in the field $\C$ of complex numbers, then

$$\lim_{q\to 1+0} (q-1) \def\z{ {\zeta}}\z_k(q) = \frac{2^{s+t}\pi^t R}{m\sqrt{|D_K|}}h,$$ where $\z_k(q)$ is Dedekind's zeta-function; $h$ is the number of divisor classes, $R$ is the regulator of $K$ (cf. Regulator of an algebraic number field) and $m$ is the number of roots of unity in $K$. By virtue of the estimate

$$|D_K|>(\frac{\pi}{4})^{2t} \frac{1}{2\pi n}e^{2n-1/6n},$$ $\lim_{n\to\infty} | D_K|=\infty$. For a quadratic field $\Q(\sqrt{d})$, where $d$ is a square-free rational integer, $d\ne 1$, one has the formulas

$$D_K = d \quad {\rm if} d\equiv 1 \pmod 4,$$

$$D_K = 4d \quad {\rm if} d\equiv 2\; {\rm or }\; 3 \pmod 4.$$ For a cyclotomic field $K=\Q(\def\e{ {\epsilon}}\e)$, where $\e$ is a primitive $p^r$-th root of unity, one has

$$D_K = \pm p^{p^{r-1} (pr-r-1)}$$ the minus sign being taken if $p^r = 4$ or $p\equiv 3$ ($\mod 4$), 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 $k$ is the field of fractions of a Dedekind ring $A$, and $K$ is a finite separable extension of $k$ of degree $n$. Let $B$ be the integral closure of the ring $A$ in $K$ and let $\def\fb{ {\mathfrak b}}\fb $ be an arbitrary fractional ideal in $B$. Then the $A$-module $D(\fb)$ generated by all discriminants of the form $D(w_1,\dots,w_n)$, where $w_1,\dots,w_n$ run through all possible bases of $K$ over $k$ and lying in $\fb$, is called the discriminant of the ideal $\fb$. $D(\fb)$ will then be a fractional ideal of $A$, and the equality $D(\fb) = N(\fb)^2 D(B)$, where $N(\fb)$ is the norm of the ideal $\fb$, is valid. The discriminant $D(B)$ is identical with the norm of the different of the ring $B$ over $A$.

V.L. Popov

Discriminant and Different of a global field

Cf. [Ja2], [La2], [We], [We2].

Let $F$ be a global field (an algebraic number field or a function field in one variable) or a local field, and let $E/F$ be a finite separable field extension. Let $A_F$ and $A_E$ be the rings of integers (principal orders) of $F$ and $E$, respectively. Let $\def\fm{ {\mathfrak m}} \fm = \{x\in E \mid \Tr(xA_E)\subset A_F$ where $\Tr : E\to F$ is the trace function.

(Let $B$ be a finite-dimensional commutative algebra over a field $k$ and $b\in B$ an element of $B$. Choose a basis $x_1,\dots,x_n$ of $B$ over $k$. Then multiplication with $b$, $b\mapsto ba$, is given by a certain matrix $M_b$. One now defines, the trace, norm and characteristic polynomial of $b$ as the trace, determinant and characteristic polynomial of the matrix $M_b$:

$$\Tr_{B/k}(b) = \Tr (M_b),\quad N_{B/k} = \det (M_b),$$

$$f_{B/k}(b)(X) = \det (X {\rm I_n} - M_b).)$$ The set $\fm$ is a fractional ideal of $A_E$. Its inverse $\fm^{-1}$ in the group of fractional ideals of the Dedekind ring $A_E$ is called the different of the field extension $E/F$, and is denoted by $\def\cD{ {\mathcal D}} \cD_{E/F}$. Sometimes (if $F\ne \Q$) it is called the relative different, and the (absolute) different of $E$ is then $\cD_{E/\Q}$. If $D/E/F$ is a tower of field extensions one has the chain theorem for differents, also called multiplicativity of differents in a tower:

$$\cD_{D/F} = \cD_{D/E} \cD_{E/F}. $$ The ideal $\cD{E/F}$ is an integral ideal of $A_E$ (i.e. $\cD_{E/F}\subset A_E$) and it is related to the discriminant $\cD(A_E)$ of the field extension $E/F$ by

$$\cD(A_E) = N_{E/F}\cD_{E/F}.$$ For the different $\cD_{E/F}$ to be divisible by a prime ideal $\def\fp{ {\mathfrak p}}\fp$ of $E$ it is necessary and sufficient that $\def\fq{ {\mathfrak q}}\fq A = \fp^e \fp_2^{e_2}\cdots \fp_m^{e_m}$ with $e>1$, where $\fq = \fp \cap A_F$. This is Dedekind's discriminant theorem. Hence a prime ideal $\fq$ of $F$ is ramified in $E/F$ if and only if $\fq$ divides the discriminant $\cD(A_E)$ of $E/F$.

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

$$L' = \{ x\in E \mid \Tr_{E/F}(xL) \subset A_F\}.$$ It is also an additive subgroup of $A_E$. Thus, the different of $E/F$ is the inverse of the complementary set of the ring of integers $A_E$ of $E$.

More generally one defines the different of an ideal $\def\fa{ {\mathfrak a}}\fa $ in $A_E$ as the inverse of its complementary set: $\cD(\fa) = (\fa')^{-1}$. It is again a (fractional) ideal of $A_E$. The different of an element $x$ in $E$ is defined as $f'(X)$ where $f'(X)$ is the derivative of the characteristic polynomial $f(X)$ of the element $x$ in $E$. If $\a\in \def\cO{ {\mathcal O}}\cO_E$, then the different $\cD(\a)$ is in $\cD_{E/F}$ and $\{1,\a,\dots,\a^{n-1}\}$ is an integral basis of $A_E$ over $A_F$ if and only if $\cD_{E/F} = \cD(\a)A_E$.

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

$$\cD_{E/F} = \prod_\fp\cD_{E_\fp/F_\fq},$$ where an ideal $\fp'$ of $E$ is identified with its completion in $E_{\fp'}$, 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 ($\cD_{E_\fp/F_\fq} = A_{E_\fp},$ for almost-all $\fp$).

Let now $R$ be a Dedekind integral domain with quotient field $F$ and let $\def\S{ {\Sigma}}\S$ be a central simple algebra over $F$ (i.e. $\S$ is a finite-dimensional associative algebra over $F$ with no ideals except $0$ and $\S$ and the centre of $\S$ is $F$). Then there is a separable normal extension $E/F$ such that $h: \S\otimes_F E \simeq M_n(E)$ (as $E$-algebras), where $M_n(E)$ is the algebra of ($n\times n$)-matrices over $E$. (Such an $E$ is called a splitting field for $\S$.) For each $x\in \S$ consider the element $h(x\otimes 1) \in M_n(E)$. The trace of this matrix is an element of $F$ (not just of $E$); it is called the reduced trace and is denoted by $\Tr : \S \to F$. (Its definition is also independent of the choice of $E$ and $h$.) Similarly one defines the reduced norm, $\def\rn{ {\rm red N}}\rn :\S\to F$, as $\rn = \det(h(x\otimes 1))$.

An $R$-lattice $\fa$ in $\S$ is an $R$-submodule of $\S$ that is finitely generated over $R$ and is such that $F_\fa = \S$. An $R$-lattice that is a subring and contains $R$ 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 $F$ not just for central simple ones.)

Let $A$ be a maximal order in $\S$. The different of $A$ in this setting is defined by $\cD(A)^{-1} = \{x\in E \mid \rn (xA)\subset R\}$. The discriminant of a central simple algebra $\S$ is the ideal $\delta = \rn(\cD(A))$. It does not depend on the choice of the maximal order $A$.

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