User:Ragnar Schroder/Sandbox: Difference between revisions

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Testing my sandbox:  
Testing my sandbox:  




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


2001: 0,40 grader
2001: 0,40 grader
Line 15: Line 19:


ÆØÅ æøå
ÆØÅ æøå
==The Galois connection==
Given a Galois group G we may look for chains <math>G = H_0  \sub H_1 \sub H_2 \ldots \sub H_n = S_1 </math> such that  <math> H_1 </math> is a normal subgroup in <math> H_0 </math>,  <math> H_2 </math> is a normal subgroup in <math> H_1 </math>, etc.
The collection of all these chains may be represented by a [[graph theory|directed graph]],  with the various subgroups as nodes and the relation <math> B \sub A </math> represented by a directed edge from A to B.
Similary,  given the collection of intermediate fields,  we may look for chains <math>K = M_n  \sub M_{n-1} \sub M_{n-2} \ldots \sub M_0 = L </math> of fields such that for all <math> i > 0 ,  M_i </math> is a normal field extension [[Galois theory glossary|(glossary)]] of <math> M_{i-1} </math>.
The collection of all these chains may be represented by a directed graph as well,  with the various fields as nodes and the relation <math> B \sub A </math> represented by a directed edge from B to A.
The Galois correspondence is an isomorphism between the two graphs.
Given a Galois group G we may look for chains <math>G = H_0  \sub H_1 \sub H_2 \ldots \sub H_n = S_1 </math> such that  <math> H_1 </math> is a normal subgroup in <math> H_0 </math>,  <math> H_2 </math> is a normal subgroup in <math> H_1 </math>, etc.
Also, given the collection of intermediate fields,  one may look for chains
<math>K = M_n  \sub M_{n-1} \sub M_{n-2} \ldots \sub M_0 = L </math> of fields such that for all <math> i > 0 ,  M_i </math> is a normal field extension [[Galois theory glossary|(glossary)]] of <math> M_{i-1} </math>.
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 <math> \lbrace H_0, ..., H_n \rbrace</math> 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 <math>K = M_n  \sub M_{n-1} \sub M_{n-2} \ldots \sub M_0 = L </math> of fields such that for all <math>i > 0 ,  M_i </math> is a normal field extension [[Galois theory glossary|(glossary)]] of <math> M_{i-1} </math>.
The Galois correspondence asserts that for each subgroup <math>H_k</math> in <math> H_1,\ldots , H_{n - 1}</math>,  there exist a corresponding fields <math>M_n</math> such that <math>M_n \sub M_k \sub M_n </math>
Given a field L that extends K, <math>K,L :  K \sub L </math>,  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 <math>x^2-5</math>, 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. <math>x^2-5 = (x-r_0)(x-r_1), r_0, r_1 \in  C</math>.  From direct inspection of the polynomial we also realize that <math>r_0 = -r_1</math>.
L = <math>\lbrace a+b r_0, a,b \in Q  \rbrace </math> is the smallest subfield of C that contains Q and both <math>r_0</math> and <math>-r_0</math>.
The are exactly 2 automorphisms of L that leave every element of Q alone: the do-nothing automorphism <math>\phi_0: a+b r_0  \rightarrow a + b r_0 </math> and the map <math>\phi_1 : a+b r_0  \rightarrow a - b r_0</math>.
Under composition of automorphisms,  these two automorphisms together form a group  isomorphic to <math>S_2</math>,  the group of permutations of two objects.
The sought for Galois group is therefore <math>S_2</math>, 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 <math>S_2</math> that there is no intermediate field extension containing Q and also roots of the polynomial.

Latest revision as of 03:34, 22 November 2023


The account of this former contributor was not re-activated after the server upgrade of March 2022.


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

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