305 -Homework set 9

[Update: I added an additional assumption to question 3.]

This set is due May 13 at 10:30 am. Remember that we will have an additional meeting that day. Details of the homework policy can be found on the syllabus and here. This set is extra credit.

Let ${X}$ be a finite set and let ${G}$ be a subgroup of the group ${S_X}$ of permutations of the elements of ${X.}$ Define a relation ${\sim}$ on ${X}$ by saying that ${x\sim y}$ iff either ${x=y}$ or ${(x,y)\in G.}$

Begin by showing that ${\sim}$ is an equivalence relation.

For each ${x\in X,}$ denote by ${E_x}$ the equivalence class of ${x,}$ i.e.,

$\displaystyle E_x=\{y\in X : x\sim y\}.$

Suppose now that in addition, for any ${x,y\in X}$ there is some permutation ${\pi\in G}$ such that ${\pi(x)=y.}$

Show that all the equivalence classes have the same size: ${|E_x|=|E_y|}$ for all ${x,y.}$

Now assume as well that ${|X|=p}$ is a prime number, and that $G$ contains at least one transposition ${(x,z)}$ for some ${x,z\in X}$ with ${x\ne z.}$

Conclude that ${G=S_X.}$

As an application, suppose that ${p}$ is a prime number and ${f\in{\mathbb Q}[x]}$ is irreducible and of degree ${{}p.}$ Assume that ${f}$ has exactly ${p-2}$ real roots and 2 complex (non-real) roots.

Conclude that

$\displaystyle {\rm Gal}({\mathbb Q}^{f(x)}/{\mathbb Q})\cong S_p.$

In particular, let ${f(x)=x^5-4x+2.}$

Show that ${{\rm Gal}({\mathbb Q}^{f(x)}/{\mathbb Q})\cong S_5.}$

Now suppose that ${2r+3}$ is prime, and let

$\displaystyle f_r(x)=(x^2+4)x(x^2-4)(x^2-16)\dots(x^2-4r^2).$

Show that if ${k}$ is an odd integer then ${|f_r(k)|\ge5.}$ Let ${g(x)=f_r(x)-2,}$ and show that ${g}$ is irreducible over ${{\mathbb Q}.}$ Now find ${{\rm Gal}({\mathbb Q}^{g(x)}/{\mathbb Q}).}$

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