Disintegration bodies

In algebra , or more precisely in body theory , a decay body is a body as small as possible in which a given polynomial is divided into linear factors . A decay field of a non-constant polynomial always exists and is uniquely determined except for isomorphism . The decay field is a normal field extension of the coefficient field of a polynomial and, if the polynomial is separable , even a Galois extension . Your Galois group is then called the Galois group of the polynomial. These terms can be generalized to any family of polynomials.

definition

Assume a field and a non-constant polynomial with coefficients . A body is called a disintegration body of (over ) if: ${\displaystyle K}$ ${\displaystyle f\in K[X]}$ ${\displaystyle K}$ ${\displaystyle L\supseteq K}$ ${\displaystyle f}$ ${\displaystyle K}$

• The polynomial is broken down into linear factors, i.e. it can be represented as ${\displaystyle f}$ ${\displaystyle L}$ ${\displaystyle f}$

${\displaystyle f=c\cdot (X-\alpha _{1})\cdot \ldots \cdot (X-\alpha _{n})}$with , and ${\displaystyle c\in K}$ ${\displaystyle \alpha _{1},\dotsc ,\alpha _{n}\in L}$

• ${\displaystyle L=K(\alpha _{1},\dotsc ,\alpha _{n})}$, that is , it is generated by adjunction of the zeros . ${\displaystyle L}$ ${\displaystyle \alpha _{1},\dotsc ,\alpha _{n}}$

If, more generally, a family of non-constant polynomials is composed , then a field is called a decay field of , if all over decay into linear factors and the field extension is generated by the zeros of . ${\displaystyle F=(f_{i})_{i\in I}}$ ${\displaystyle K[X]}$ ${\displaystyle L\supseteq K}$ ${\displaystyle F}$ ${\displaystyle f_{i}}$ ${\displaystyle L}$ ${\displaystyle L/K}$ ${\displaystyle f_{i}}$

Existence and uniqueness

If, for example, a polynomial with rational coefficients, then the existence of a decay field is easy to show: According to the fundamental theorem of algebra , the polynomial in the field of complex numbers breaks down into linear factors. By the addition of all complex zeros of , one obtains a disintegration body of over . This approach can be generalized: With the help of Zorn's lemma , it can be shown that for any field there is an extension field that is algebraically closed , for example the algebraic closure of . If an arbitrary family of polynomials is in , then each over decays into linear factors. The intersection of all subfields of that contain and in which all decay into linear factors is then the smallest expansion field of that contains all zeros of the polynomials , i.e. a decay field of the family . ${\displaystyle f\in \mathbb {Q} [X]}$ ${\displaystyle f}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \alpha _{1},\dotsc ,\alpha _{n}}$ ${\displaystyle f}$ ${\displaystyle \mathbb {Q} (\alpha _{1},\dotsc ,\alpha _{n})}$ ${\displaystyle f}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle K}$ ${\displaystyle A}$ ${\displaystyle {\overline {K}}}$ ${\displaystyle K}$ ${\displaystyle F}$ ${\displaystyle K[X]}$ ${\displaystyle f\in F}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle f\in F}$ ${\displaystyle K}$ ${\displaystyle f\in F}$ ${\displaystyle F}$

The disintegration body of a family is uniquely determined except for -isomorphism. This means: Are and two splitting from over , then there is a Körperisomorphismus with for all . ${\displaystyle F\subseteq K[X]}$ ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle L'}$ ${\displaystyle F}$ ${\displaystyle K}$ ${\displaystyle \varphi \colon L\to L'}$ ${\displaystyle \varphi (x)=x}$ ${\displaystyle x\in K}$

construction

The existence of a decay field of a polynomial can also be shown by a direct construction without Zorn's lemma. What is essential here is the statement that for every non-constant polynomial there is a field in which there is a zero. According to an idea by Leopold Kronecker ( Kronecker's theorem ), such a body can be constructed in the following way: Let it be an irreducible factor of . Then the main ideal generated by is a maximal ideal in and hence the factor ring is a solid. For the element ${\displaystyle f\in K[X]}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle (g)=gK[X]}$ ${\displaystyle K[X]}$ ${\displaystyle K[X]/(g)}$

${\displaystyle {\overline {X}}=X+(g)\in K[X]/(g)}$

applies

${\displaystyle g({\overline {X}})=g(X)+(g)=0+(g)={\overline {0}}}$,

that is , a zero of and therefore also of . ${\displaystyle {\overline {X}}}$ ${\displaystyle g}$ ${\displaystyle f}$

The existence of a disintegration body of can now easily be shown with complete induction on the degree of : ${\displaystyle f\in K[X]}$ ${\displaystyle n}$ ${\displaystyle f}$

• For the start of induction there is even a disintegration body of . ${\displaystyle n=1}$ ${\displaystyle K}$ ${\displaystyle f}$
• For there after what is shown above an extension field of in which a zero has. In can be decomposed as with a polynomial of degree . According to the induction assumption, has the zeros in a disintegration body. This is a disintegration body of . ${\displaystyle n\geq 2}$ ${\displaystyle K'}$ ${\displaystyle K}$ ${\displaystyle f}$ ${\displaystyle \alpha }$ ${\displaystyle K'}$ ${\displaystyle f}$ ${\displaystyle f=(X-\alpha )\cdot h}$ ${\displaystyle h\in K'[X]}$ ${\displaystyle n-1}$ ${\displaystyle h}$ ${\displaystyle \alpha _{1},\dotsc ,\alpha _{n-1}}$ ${\displaystyle K(\alpha ,\alpha _{1},\dotsc ,\alpha _{n-1})}$ ${\displaystyle f}$

properties

• The decay field of a family is minimal in the following sense: If there is a field with such that every polynomial over decays into linear factors, then applies . ${\displaystyle L}$ ${\displaystyle F\subseteq K[X]}$ ${\displaystyle Z}$ ${\displaystyle K\subseteq Z\subseteq L}$ ${\displaystyle f\in F}$ ${\displaystyle Z}$ ${\displaystyle Z=L}$

• The decay field of a finite set of polynomials in is equal to the decay field of the product polynomial . ${\displaystyle \{f_{1},\dotsc ,f_{k}\}}$ ${\displaystyle K[X]}$ ${\displaystyle f=f_{1}\cdot \ldots \cdot f_{k}}$

• The degree of expansion of the decay field of a polynomial of degree is a factor of , in particular, applies . If over is irreducible , then . ${\displaystyle [L:K]}$ ${\displaystyle f\in K[X]}$ ${\displaystyle n}$ ${\displaystyle n!}$ ${\displaystyle [L:K]\leq n!}$ ${\displaystyle f}$ ${\displaystyle K}$ ${\displaystyle [L:K]\geq n}$

• If the disintegration body is a family , then the body expansion is algebraic and normal . If all are separable , then there is a separable extension, i.e. even a Galois extension . ${\displaystyle L}$ ${\displaystyle F\subseteq K[X]}$ ${\displaystyle L/K}$ ${\displaystyle f\in F}$ ${\displaystyle L/K}$

Examples

• If a polynomial already decays into linear factors, then trivially the decay field of . Therefore, for example, the polynomials , or from, all have themselves as decay fields. ${\displaystyle f\in K[X]}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle f}$ ${\displaystyle X-3}$ ${\displaystyle X^{2}-4=(X-2)(X+2)}$ ${\displaystyle (X+1)^{5}}$ ${\displaystyle \mathbb {Q} [X]}$ ${\displaystyle \mathbb {Q} }$

• The polynomial is divided into into linear factors: . So the disintegration body of is . ${\displaystyle f=X^{2}-2\in \mathbb {Q} [X]}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle f=(X-{\sqrt {2}})(X+{\sqrt {2}})}$ ${\displaystyle f}$ ${\displaystyle \mathbb {Q} ({\sqrt {2}},-{\sqrt {2}})=\mathbb {Q} ({\sqrt {2}})}$

• The disintegration body of with the complex zeros and the body are analogous . ${\displaystyle g=X^{2}+1\in \mathbb {Q} [X]}$ ${\displaystyle \mathrm {i} }$ ${\displaystyle \mathrm {-i} }$ ${\displaystyle \mathbb {Q} (\mathrm {i} )}$

• The disintegration body of is therefore . ${\displaystyle (X^{2}-2)(X^{2}+1)\in \mathbb {Q} [X]}$ ${\displaystyle \mathbb {Q} ({\sqrt {2}},\mathrm {i} )}$

• The polynomial understood as a polynomial with real coefficients, i.e. as an element of , has as a decay field. This shows that the specification of the coefficient field of a polynomial is essential for the determination of its decay field. ${\displaystyle X^{2}+1}$ ${\displaystyle \mathbb {R} [X]}$ ${\displaystyle \mathbb {R} (\mathrm {i} )=\mathbb {C} }$

• The polynomial has a zero in the body , but this body is not the decay field of , because the other two zeros and in are non-real, so they cannot lie in the real subfield . The disintegration body of is . ${\displaystyle h=X^{3}-2\in \mathbb {Q} [X]}$ ${\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}})}$ ${\displaystyle h}$ ${\displaystyle {\sqrt[{3}]{2}}e^{2\pi \mathrm {i} /3}}$ ${\displaystyle {\sqrt[{3}]{2}}e^{4\pi \mathrm {i} /3}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}})\subseteq \mathbb {R} }$ ${\displaystyle h}$ ${\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}},e^{2\pi \mathrm {i} /3})}$

Applications

In Galois theory , the zeros of a polynomial are examined with the aid of its decay field . A group , the Galois group , is assigned to the body extension . The group is called the Galois group of the polynomial . According to the main theorem of Galois theory, the subgroups of clearly correspond to the intermediate bodies . In this way, numerous classic problems of algebra can be solved, such as the question of which numbers can be constructed with compasses and ruler or which polynomial equations can be solved by radicals (see e.g. Abel-Ruffini's theorem ). ${\displaystyle f\in K[X]}$ ${\displaystyle L}$ ${\displaystyle L/K}$ ${\displaystyle \operatorname {Gal} (L/K)}$ ${\displaystyle f}$ ${\displaystyle \operatorname {Gal} (L/K)}$ ${\displaystyle Z}$ ${\displaystyle K\subseteq Z\subseteq L}$

The circular dividing bodies are special disintegrating bodies : The complex solutions of the equation with are the -th roots of unity for . The -th field of circular division is because of the decay fields of the polynomial . ${\displaystyle z^{n}=1}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n}$ ${\displaystyle \zeta _{n,k}=e^{2\pi \mathrm {i} k/n}}$ ${\displaystyle k=0,\dotsc ,n-1}$ ${\displaystyle n}$ ${\displaystyle \mathbb {Q} (e^{2\pi \mathrm {i} /n})}$ ${\displaystyle \zeta _{k,n}=(\zeta _{n,1})^{k}}$ ${\displaystyle X^{n}-1\in \mathbb {Q} [X]}$

The finite fields can also be represented as decay fields: If a prime number , then the remainder class ring is a field and is denoted by. For a natural number , the polynomial has exactly different zeros in an algebraic closure . The disintegration body of is then a body with elements. One can show that in this way all finite bodies can be created. ${\displaystyle p\in \mathbb {N} }$ ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle \mathbb {F} _{p}}$ ${\displaystyle n}$ ${\displaystyle f=X^{p^{n}}-X\in \mathbb {F} _{p}[X]}$ ${\displaystyle p^{n}}$ ${\displaystyle f}$ ${\displaystyle \mathbb {F} _{p^{n}}}$ ${\displaystyle p^{n}}$

literature

• Siegfried Bosch : Algebra. 8th edition. Springer Spectrum, Berlin / Heidelberg 2013, ISBN 978-3-642-39566-6 , Section 3.5: Disintegration bodies .
• Christian Karpfinger, Kurt Meyberg: Algebra: Groups - Rings - Body. 3. Edition. Springer Spectrum, Berlin / Heidelberg 2013, ISBN 978-3-8274-3011-3 , Section 24.2: Disintegration bodies .
• Kurt Meyberg: Algebra, Part 2 : Carl Hanser Verlag (1976), ISBN 3-446-12172-2 , Section 6.5: Disintegration bodies .