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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