User:Ragnar Schroder/Sandbox

Testing my sandbox:

1995: 0,28 grader 1997: 0,36 grader 1998: 0,52 grader

2001: 0,40 grader 2002: 0,46 grader 2003: 0,46 grader 2004: 0,43 grader 2005: 0,48 grader 2006: 0,42 grader 2007: 0,41 grader

ÆØÅ æøå

The Galois connection

Given a Galois group G we may look for chains $G=H_{0}\subset H_{1}\subset H_{2}\ldots \subset H_{n}=S_{1}$ such that $H_{1}$ is a normal subgroup in $H_{0}$, $H_{2}$ is a normal subgroup in $H_{1}$, etc.

The collection of all these chains may be represented by a directed graph, with the various subgroups as nodes and the relation B is a normal subgroup in A represented by a directed edge from A to B.

Similary, given the collection of intermediate fields, we may look for chains $K=M_{n}\subset M_{n-1}\subset M_{n-2}\ldots \subset M_{0}=L$ of fields such that for all $i>0,M_{i}$ is a normal field extension (glossary) of $M_{i-1}$.

The collection of all these chains may be represented by a directed graph as well, with the various fields as nodes and the relation A is a normal extension of B represented by a directed edge from A to B.

The Galois correspondence is an $\phi$ between the two graphs.

Given a Galois group G we may look for chains $G=H_{0}\subset H_{1}\subset H_{2}\ldots \subset H_{n}=S_{1}$ such that $H_{1}$ is a normal subgroup in $H_{0}$, $H_{2}$ is a normal subgroup in $H_{1}$, etc.

Also, given the collection of intermediate fields, one may look for chains $K=M_{n}\subset M_{n-1}\subset M_{n-2}\ldots \subset M_{0}=L$ of fields such that for all $i>0,M_{i}$ is a normal field extension (glossary) of $M_{i-1}$.

The Galois connection is a bijective function f from all subgroups in G participating in any chain of normal subgroups to all intermediate fields participating in any chain of normal field extensions.

Given any such chain, the Galois correspondence is a function f from the set $\lbrace H_{0},...,H_{n}\rbrace$ to the set

The Galois correspondence is a function f from any such element in any such chain such that

The Galois correspondence asserts that for each such chain of normal subgroups, there exists a corresponding chain $K=M_{n}\subset M_{n-1}\subset M_{n-2}\ldots \subset M_{0}=L$ of fields such that for all $i>0,M_{i}$ is a normal field extension (glossary) of $M_{i-1}$.

The Galois correspondence asserts that for each subgroup $H_{k}$ in $H_{1},\ldots ,H_{n-1}$, there exist a corresponding fields $M_{n}$ such that $M_{n}\subset M_{k}\subset M_{n}$

Given a field L that extends K, $K,L:K\subset L$, the field extension and

If L is a splitting field for some polynomial with coefficients over K, then L:K is a normal and finite field extension.

If

The Galois group of a polynomial - a basic example

As an example, let us look at the second-degree polynomial $x^{2}-5$, with the coefficients {-5,0,1} viewed as elements of Q.

This polynomial has no roots in Q. However, from the fundamental theorem of algebra we know that it has exacly two roots in C, and can be written as the product of two first-degree polynomials there - i.e. $x^{2}-5=(x-r_{0})(x-r_{1}),r_{0},r_{1}\in C$. From direct inspection of the polynomial we also realize that $r_{0}=-r_{1}$.

L = $\lbrace a+br_{0},a,b\in Q\rbrace$ is the smallest subfield of C that contains Q and both $r_{0}$ and $-r_{0}$.

The are exactly 2 automorphisms of L that leave every element of Q alone: the do-nothing automorphism $\phi _{0}:a+br_{0}\rightarrow a+br_{0}$ and the map $\phi _{1}:a+br_{0}\rightarrow a-br_{0}$.

Under composition of automorphisms, these two automorphisms together form a group isomorphic to $S_{2}$, the group of permutations of two objects.

The sought for Galois group is therefore $S_{2}$, which has no nontrivial subgroups.

The 3 requirements (explained below) for the Galois correspondence to exist happen to be fullfilled, so we we conclude from the subgroup structure of $S_{2}$ that there is no intermediate field extension containing Q and also roots of the polynomial.