Talk:Galois theory: Difference between revisions
imported>Ragnar Schroder (New page: {{subpages | abc = Galois theory | cat1 = Mathematics | cat2 = | cat3 = | cat_check = yes | status = 3...) |
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The following is just a scratch to work out the 1st non-stub version of the article | |||
'''Galois theory''' is an area of mathematical study that originated with [[Evariste Galois]] around 1830, as part of an effort to understand the relationships between the roots of [[polynomial|polynomials]], in particular why there are no simple formulas for extracting the roots of the general polynomial of fifth (or higher) degree. | |||
==Introduction== | |||
Galois expressed his theory in terms of polynomials and [[complex number|complex numbers]], today Galois theory is usually formulated using general [[field theory]]. | |||
Key concepts are [[Field extension|field extensions]] and [[Group theory|groups]], which should be thoroughly understood before Galois theory can be properly studied. | |||
The core idea behind Galois theory is that given a polynomial <math>\alpha</math> with coefficients in a field K (typically the rational numbers), there exists | |||
*a field L that contains K (or a field [[isomorphic]] to K) as a subfield, and also the roots of <math>\alpha</math>. | |||
*a group containing all [[automorphisms]] in L that leave the elements in K untouched. | |||
Providing certain technicalities are fullfilled, the structure of this group contains information about the nature of the roots, and whether the equation <math>\alpha = 0</math> has solutions expressible as radical expressions - i.e. formulas involving a simple sequence of ordinary arithmetical expressions and rational powers. | |||
==Summary of the theory== | |||
===Basic concepts/glossary=== | |||
====Field extensions==== | |||
Any field K can be "extended" by including one or more "foreign" elements, f.i. the field Q can be extended by including sqr(2). The resulting field is the subset of R described by a+b sqrt(2), a,b in Q. | |||
Similarly, if r1, r2, ... rn are roots of a polynomial α , a lattice of extension fields may be constructed. ... | |||
Algebraic extension vs transcendental... | |||
The order of an extension ... | |||
Normal extensions and splitting fields ... | |||
Given a polynomial <math>\alpha</math> with coefficients in a field K, there exists a field M ⊇ K - known as a ''splitting field'' of <math>\alpha</math> - which contains all the roots of <math>\alpha</math>. | |||
===The Galois correspondence=== | |||
The correspondence between the Galois group subgroup structure and the field extension lattice ... | |||
Soluble groups ... Why neither the quintic nor its friend S5 are "soluble". | |||
Caveat - separability - only relevant with non-zero characteristic fields. | |||
:How much to rely on external "Field extensions" article? | |||
:[[User:Ragnar Schroder|Ragnar Schroder]] 05:38, 12 December 2007 (CST) | |||
Revision as of 05:38, 12 December 2007
The following is just a scratch to work out the 1st non-stub version of the article
Galois theory is an area of mathematical study that originated with Evariste Galois around 1830, as part of an effort to understand the relationships between the roots of polynomials, in particular why there are no simple formulas for extracting the roots of the general polynomial of fifth (or higher) degree.
Introduction
Galois expressed his theory in terms of polynomials and complex numbers, today Galois theory is usually formulated using general field theory.
Key concepts are field extensions and groups, which should be thoroughly understood before Galois theory can be properly studied.
The core idea behind Galois theory is that given a polynomial with coefficients in a field K (typically the rational numbers), there exists
- a field L that contains K (or a field isomorphic to K) as a subfield, and also the roots of .
- a group containing all automorphisms in L that leave the elements in K untouched.
Providing certain technicalities are fullfilled, the structure of this group contains information about the nature of the roots, and whether the equation has solutions expressible as radical expressions - i.e. formulas involving a simple sequence of ordinary arithmetical expressions and rational powers.
Summary of the theory
Basic concepts/glossary
Field extensions
Any field K can be "extended" by including one or more "foreign" elements, f.i. the field Q can be extended by including sqr(2). The resulting field is the subset of R described by a+b sqrt(2), a,b in Q.
Similarly, if r1, r2, ... rn are roots of a polynomial α , a lattice of extension fields may be constructed. ...
Algebraic extension vs transcendental...
The order of an extension ...
Normal extensions and splitting fields ...
Given a polynomial with coefficients in a field K, there exists a field M ⊇ K - known as a splitting field of - which contains all the roots of .
The Galois correspondence
The correspondence between the Galois group subgroup structure and the field extension lattice ...
Soluble groups ... Why neither the quintic nor its friend S5 are "soluble".
Caveat - separability - only relevant with non-zero characteristic fields.
- How much to rely on external "Field extensions" article?
- Ragnar Schroder 05:38, 12 December 2007 (CST)