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 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 $r_{0}=-r_{1}$.

We now look for the smallest subfield of C that contains Q and both $r_{0}$ and $-r_{0}$, which is L = { $a+br_{0},a,b\in Q$ }. Since $r_{0}^{2}=5\in Q$, all products and sums are well defined. This field is then the smallest extension of Q by the roots of $\alpha$.

Now, in order to find the Galois group, we need to look at all possible automorphisms of L that leave every element of Q alone.

The only such automorphisms are the null automorphism and the map $a+br_{0}\rightarrow a-br_{0}$.

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

The sought for Galois group is therefore $S_{2}$.