Talk:Galois theory: Difference between revisions

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Definition: Algebra concerned with the relation between solutions of a polynomial equation and the fields containing those solutions. [d] [e]

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The following is just a scratch to work out the 1st non-stub version of the article

Text from below here is not in the article yet

However, we may create an extension field L containing two elements $r_{0},r_{1}$ such that $x^{2}-5=(x-r_{0})(x-r_{1})$ . By the fundamental theorem of algebra this is always possible - there exists a subfield L of C such that $K\subset L\subseteq C$

As an example, the second-degree polynomial $x^{2}-5$ - when the coefficients {0,1,5} are viewed as elements of Q - turns out to have the Galois group $S_{2}$ .

From the subgroup structure of $S_{2}$ - the only proper subgroup is the trivial group $S_{1}$ - we may conclude that the chain of extension fields from Q to the smallet extension field of Q such that the polynomial splits is trivial - no intermediate extension fields exist.

Finding the Galois group of a polynomial is in general a tedious process, in this example it was easy, since the group had to be contained in $S_{2}$ .

Looking again at the polynomial $x^{2}-5$ , one may wonder exactly what it's "Galois group" is, and how to find it.

... Mention something about the Fundamental theorem of algebra, which implies that there is a subfield in C such that $x^{2}-5$ can be split into linear factors $(x-r_{1})(x-r_{2}),r_{1},r_{2}\subset C$ ...

...Mention $S_{n}$ , where n is the degree of the polynomial ...

Basic concepts/glossary

Polynomial over a field K: An expression of the form $a_{n-1}x^{n-1}+...+a_{1}x^{1}+a_{0}$ , with $a_{0},a_{1},...a_{n-1}\in K$ .
Root of a polynomial $\alpha$ : a number r such that $\alpha (r)=0$
A splitting field for a polynomial $\alpha$ : A field which contains the original field K as a subfield, and also contains all the roots of $\alpha$ .

Summary of the theory

Given a polynomial $\alpha$ with coefficients in some field K, it may be the case that the equation $\alpha =0$ has no solutions in K. In that case, $\alpha$ is said to be irreducible in K.

Anyway, if K is a subfield of C, we are guaranteed by the fundamental theorem of algebra that there exists a subfield of C containing K and all the roots.

...blabber about field of characteristic <> 0 ...

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 $\alpha$ with coefficients in a field K, there exists a field M ⊇ K - known as a splitting field of $\alpha$ - which contains all the roots of $\alpha$ .

The Galois correspondence

The correspondence between the Galois group subgroup structure and the field extension lattice ...

Caveat - separability - only relevant with non-zero characteristic fields.

Soluble groups ... Why neither the quintic nor its friend S5 are "soluble". Why 60 degree angles won't let themselves be "trisected". Why this was a triumph for Galois theory, 2000+ year old riddles solved.

How much to rely on an extra "Field extensions" article?

Ragnar Schroder 05:38, 12 December 2007 (CST)

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-'''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 smallest possible field L that contains K (or a field [[isomorphic]] to K) as a subfield and also all the roots of <math>\alpha</math>.  This field is known as the extension of K by the roots of <math>\alpha</math>.
-*a group containing all [[automorphisms]] in L that leave the elements in K untouched - the Galois group of the polynomial <math>\alpha</math>.
-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 a finite formula involving   only ordinary arithmetical operations (addition, subtraction, multiplication, division and rational powers) on the coefficients.
-===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>.
-We now look for the smallest subfield of C that contains Q and both <math>r_0</math> and <math>-r_0</math>, which is L = { <math>   a+b r_0, a,b \in Q  </math>  }.  Since <math>r_0^2 = 5 \in Q</math>,  all products and sums are well defined.  This field is then the smallest extension of Q by the roots of <math>\alpha</math>.
-Now, in order to find the Galois group,  we need to look at all possible automorphisms of L that leave every elements of Q alone.
-The only such automorphisms are the null automorphism and the map <math>a+b r_0   \rightarrow a - b r_0</math>.
-Under composition of automorphisms,  these two automorphisms together are isomorphic to the group <math>S_2</math>,  the group of permutations of two objects.
-The sought for Galois group is therefore <math>S_2</math>.

Talk:Galois theory: Difference between revisions

Revision as of 04:04, 19 December 2007