Galois Theory and Splitting Fields

Question

So I have an exam tomorrow and I think I'm rather prepared as far as the theory goes (I have the theorems in the book memorized, etc), but I am rather worried about any "concrete" questions I may get (by this I mean as concrete as Abstract Algebra goes). We are covering ring theory up through Galois theory and just touching on Solvability.

Essentially I'm asking if people have any nice tricks/methods/theorems that are easily overlooked/forgotten, if people had any good practice problems I could work over tomorrow morning to get warmed up for the exam, and lastly how do you go about tackling questions like:

*Find the degree of an extension field over a given polynomial over $\mathbb{Q}$.

*Determine the Galois group of a given polynomial over $\mathbb{Q}$

*etc

Thank you

Answer 1

I know two methods (at the level you seem to be) that are usually used in Galois theory that you should check, there are also methods to show that a polynomial (over $\mathbb{Z}$) is irreducible.

• Showing irreducibility. For a polynomial $P$ written as :

$$P=a_nX^n+a_{n-1}X^{n-1}+...+a_0 $$

Then the first method is Eisenstein criterion, if there exists a prime $p$ such that $p$ divides $a_0,...,a_{n-1}$, $p^2$ does not divide $a_0$ and $p$ does not divide $a_n$ then $P$ is irreducible.

On the other hand,if $a_0,...,a_n\in\mathbb{Z}$ then if there exists some $p$ such that $P$ mod $p$ is irreducible as a polynomial in $\mathbb{F}_p[X]$ then $P$ is irreducible in $\mathbb{Q}[X]$.

• The antisymmetric group. Once you have shown that a polynomial $P$ of degree $n$ is irreducible (let's say that its roots are $x_1,...,x_n$) then you might want to know if its Galois group is included in $\mathfrak{A}_n$ or not. To know this you just have to calculate :

$$Discr(P):=Res(P,P')=\prod_{i\neq j}(x_i-x_j)$$

Then you can show that the Galois group over $\mathbb{Q}$ of $P$ is included in $\mathfrak{A}_n$ if and only if :

$$\prod_{i< j}(x_i-x_j)\in\mathbb{Q} $$

if and only if $Discr(P)=q^2$ for some $q\in\mathbb{Q}$.

For this method to be usefull you need to know that there are formula for discriminant which are polynomials in coefficients of $P$ (google it to know about $Res(P,P')$).

• The degree five method. Take $P$ an irreducible polynomial of degree five. Then from the theory you know that its Galois group $G$ acts transitively on the set of five roots of $P$, from this you know that the order of $G$ is a multiple of $5$ and by Cauchy theorem contains an element of order $5$, because you are in $\mathfrak{S}_5$ this element cannot but be a $5$-cycle.

Now, assume that you can show that $P$ has exactly $3$ real roots (by using intermediate value theorem for instance) then it has two complex non-real roots which cannot but be conjugate to each other. Hence the complex conjugation will induce an automorphism permuting two roots and fixing the three other, that is complex conjugation will induce a transposition. Now it is a good exercise in group theory to show that $\mathfrak{S}_p$ for $p$ prime is generated by any couple $(\tau,c)$ where $\tau$ is a transposition and $c$ is a $p$-cycle. This shows that $G$ is necessarily $\mathfrak{S}_5$. Exercise : show that this polynomials are irreducible and have Galois group $\mathfrak{S}_5$ :

$$P_1=X^5-150X+6$$

$$P_2=X^5-35X^2+3$$