Talk:Galois theory

Basic summary of Galois theory

The core idea behind Galois theory is that given a polynomial $\alpha$ 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 $\alpha$. This field is known as the extension of K by the roots of $\alpha$.

• a group containing all automorphisms in L that leave the elements in K untouched - the Galois group of the polynomial $\alpha$.

Providing certain technicalities are fullfilled, the structure of this group contains information about the nature of the roots, and whether the equation $\alpha =0$ has solutions expressible as a finite formula involving only ordinary arithmetical operations (addition, subtraction, multiplication, division and rational powers) on the coefficients.

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)