Discriminant (algebraic number theory)

In algebraic number theory , the discriminant designates a main ideal in a wholeness ring , which makes a number-theoretical statement about the field expansion of two number fields.

definition

Let be a ring , a sub-ring such that a free - module is of rank . For is called the discriminant of . ${\displaystyle B}$ ${\displaystyle A\subseteq B}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle n\;(n\in \mathbb {N} )}$ ${\displaystyle (x_{1},x_{2},\dots ,x_{n})\in B^{n}}$ ${\displaystyle D(x_{1},x_{2},\dots ,x_{n}):=\det \left(\mathrm {Tr} _{B/A}(x_{i}\cdot x_{j})_{i,j}\right)\in A}$ ${\displaystyle (x_{1},x_{2},\dots ,x_{n})}$

If a- base of represents, then the discriminant is uniquely determined up to one unit in , in particular the main ideal generated by in is independent of the base choice. This main ideal is denoted by and is called the discriminant of over . ${\displaystyle (x_{1},x_{2},\dots ,x_{n})}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle D(x_{1},x_{2},\dots ,x_{n})}$ ${\displaystyle A}$ ${\displaystyle {\mathfrak {D}}_{B/A}}$ ${\displaystyle B}$ ${\displaystyle A}$

Properties and application

• Let be a field of characteristic , a field extension of of degree and the various - algebra monomorphisms of in the algebraic closure. Then for a base of : ${\displaystyle K}$ ${\displaystyle 0}$ ${\displaystyle L}$ ${\displaystyle K}$ ${\displaystyle n\;(n\in \mathbb {N} )}$ ${\displaystyle \sigma _{1},\sigma _{2},\dots ,\sigma _{n}}$ ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle K}$ ${\displaystyle (x_{1},x_{2},\dots ,x_{n})}$ ${\displaystyle L}$

${\displaystyle D(x_{1},x_{2},\dots ,x_{n})=\det \left((\sigma _{i}(x_{j}))_{i,j}\right)^{2}\neq 0}$

• Let be two number fields, with the associated wholeness rings . Then the following holds for a prime ideal : is branched if and only if holds. In particular it follows from this that there are only finitely many branched prime ideals (unambiguous prime decomposition of , cf. Dedekindring ). ${\displaystyle K\subseteq L}$ ${\displaystyle A\subseteq B}$ ${\displaystyle {\mathfrak {p}}\subseteq A}$ ${\displaystyle {\mathfrak {p}}\subseteq B}$ ${\displaystyle {\mathfrak {p}}\supseteq {\mathfrak {D}}_{B/A}}$ ${\displaystyle {\mathfrak {D}}_{B/A}}$

example

Be ; denote the equivalence class of in . ${\displaystyle A:=\mathbb {Q} ,\;B:=\mathbb {Q} [X]/(X^{2}+bX+c),\quad b,c\in \mathbb {Q} }$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle B}$

So what corresponds to the discriminant of the polynomial . ${\displaystyle D_{B/A}(1,x)=\det {\begin{pmatrix}\mathrm {Tr} _{B/A}(1)&\mathrm {Tr} _{B/A}(x)\\\mathrm {Tr} _{B/A}(x)&\mathrm {Tr} _{B/A}(x^{2})\end{pmatrix}}=\det {\begin{pmatrix}2&-b\\-b&b^{2}-2c\end{pmatrix}}=b^{2}-4c}$ ${\displaystyle X^{2}+bX+c}$

To calculate the tracks used :

${\displaystyle \mathrm {Tr} _{B/A}(1)=\mathrm {Tr} _{B/A}{\begin{pmatrix}1&0\\0&1\end{pmatrix}}=2}$

${\displaystyle \mathrm {Tr} _{B/A}(x)=\mathrm {Tr} _{B/A}{\begin{pmatrix}0&-c\\1&-b\end{pmatrix}}=-b}$

${\displaystyle \mathrm {Tr} _{B/A}(x^{2})=\mathrm {Tr} _{B/A}(-b\cdot x-c)=-b\cdot \mathrm {Tr} _{B/A}(x)-c\cdot \mathrm {Tr} _{B/A}(1)=b^{2}-2c}$

Discriminant of a number field

Let K be a number field and O _{K be} its entirety ring. Let b ₁ , ..., b _{n be} a basis of O _K as Z -module, and let {σ ₁ , ..., σ _n } be the embeddings of K in the complex numbers. The discriminant of K is the square of the determinant of the n -mal- n - matrix B whose ( i , j ) entry is σ _i ( b _j ).

${\displaystyle \Delta _{K}=\left(\operatorname {det} \left({\begin{array}{cccc}\sigma _{1}(b_{1})&\sigma _{1}(b_{2})&\cdots &\sigma _{1}(b_{n})\\\sigma _{2}(b_{1})&\ddots &&\vdots \\\vdots &&\ddots &\vdots \\\sigma _{n}(b_{1})&\cdots &\cdots &\sigma _{n}(b_{n})\end{array}}\right)\right)^{2}.}$

See also

Discriminant