189-457B: Algebra 4 Assignment 4

Due: Wednesday, March 22

1. Does every extension K of degree 4 over a field F contain a sub-extension of degree 2 over F? If yes, prove your statement, and if no, give a counterexample.

The following series of exercises is meant to get you to prove that every finite group occurs as the Galois group of some finite extension K/F, where F is itsef a finite extension of Q.

2. Let p be a prime number and let S_{p} denote the group of permutations on p elements. Show that any subgroup G ⊂ S_{p} that contains a transposition (i.e., a permutation that interchanges two elements and leaves all others fixed) and a permutation of order p is necessarily equal to S_{p}. Show that the same conclusion holds if G contains a transposition and acts transitively on {1, 2, …, p}. Finally, show that this conclusion is false if the prime p is replaced by an even integer > 2.

3. Using the result in part 2, show that any irreducible polynomial in Q[x] of degree p having exactly p − 2 real roots has Galois group S_{p} over Q (i.e., the automorphism group of its splitting field acts on the roots of this polynomial as the full permutation group on p elements.)

4. If f(x) ∈ R[x] is any polynomial having exactly k distinct real roots, show that there exists ϵ > 0 for which f(x) + a has exactly k real roots, for all a ∈ R with |a| < ϵ. Give an example to show that the assumption that the roots of f(x) are distinct is essential for the conclusion to hold.

5. For any n, show that there is an irreducible polynomial in Q[x] of degree n having exactly n − 2 real roots. (Hint: starting with any polynomial f(x) ∈ Q[x] with exactly n−2 distinct real roots, exercise 4 shows that f(x)+a has the same property for infinitely many a ∈ Q. Now, make a judicious choice of f(x) ∈ Z[x] and a ∈ Q for which the Eisenstein irreducibility criterion can be applied.)

6. Show that every finite groupcan be realised as a subgroup of S_{p} for a large enough prime p. Conclude from this and from what you have done in exercises 2-5 that every finite group is the Galois group of an extension K/F, where F is itself a finite extension of Q.

Remark: It is natural to ask whether any finite group can occur as the Galois group of a finite extension of Q. This is widely believed to be the case, but a proof is not known. The problem of realising any finite group as a Galois group of an extension of Q is known as the “inverse problem of Galois theory”, and is one of the most famous open problems in the subject.

7. Show that any sequence of elements x_{i} ∈ Q̄ (or in any field, for that matter) that satisfies the recursion

x_{i−1}x_{i+1} = x_{i} + 1, for all i ≥ 1

is necessarily periodic of period 5. If (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) is the basic period in such a sequence, show that the 5 elementary symmetric functions of x_{1}, …, x_{5} can be expressed in terms of the quantities

s := x_{1} + ⋯ + x_{5}, t = x_{1}x_{2} + x_{2}x_{3} + x_{3}x_{4} + x_{4}x_{5} + x_{5}x_{1}

by the formulae

x_{1} + ⋯ x_{2} = s; x_{1}x_{2} + x_{1}x_{3} + ⋯ x_{4}x_{5} = s + t + 5;
x_{1}x_{2}x_{3} + ⋯ x_{1}x_{2}x_{4} + ⋯ + x_{3}x_{4}x_{5} = s^{2} + s − 2t − 5;

x_{1}x_{2}x_{3}x_{4} + ⋯ + x_{2}x_{3}x_{4}x_{5} = 2s + t + 5; x_{1}x_{2}x_{3}x_{4}x_{5} = s + 3.

8. Using exercise 7, show that the degree 5 polynomial

x^{5} − sx^{4} + (s + t + 5)x^{3} − (s^{2} + s − 2t − 5)x^{2} + (2s + t + 5)x − (s + 3)

with coefficients in the field Q(s, t) of rational functions in the two indeterminates s and t has Galois group contained in the dihedral group D_{10} of cardinality 10.

9. Experimental exercise. This exercise assumes some familiarity with a computer algebra package like Pari/GP, which is available for free on the internet. Enter the three-variable polynomial

       f = x^{5} − s ∗ x^{4} + (s + t + 5) ∗ x^{3} − (s^{2} + s − 2 ∗ t − 5) ∗ x^{2}

+(2 ∗ s + t + 5) ∗ x − (s + 3)

in the Pari command line, and ask Pari to calculate the Galois group of the specialised polynomial as the parameters s and t range over all integer values between 1 and 10, by typing something like

       for(s = 1, 10, for(t = 1, 10, print(polgalois(eval(f)))))

What do you observe?