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MA 3419: Galois theory Homework problems due October 20, 2015 Solutions to this are due by the end of the 4pm class on Tuesday October 20. Please attach a cover sheet with a declaration http://tcd-ie.libguides.com/plagiarism/declaration confirming that you know and understand College rules on plagiarism. Please put your name and student number on each of the sheets you are handing in. (Or, ideally, staple them together). 1. (a) Give an example of a non-Abelian group G for which G/Z(G) is Abelian. (As we know from class, it cannot be cyclic). (b) Let R = Z/4Z. Give an example of a polynomial x2 + ax + b ∈ R[x] which has three distinct roots in R. (c) Show that the equation x2 + 1 = 0 has infinitely many solutions in the (noncommutative) ring H of quaternions. 2. (a) Show that if in a group G we have g2 = e for all g, then G is Abelian (e is the unit element). (b) Show that a non-Abelian group G of order 6 must have an element of order 3. Use that to demonstrate that every non-Abelian group of order 6 is isomorphic to S3 . [Hint: if g ∈ G is of order 3, and h ∈ G is distinct from e, g, g2 , show that h2 = e, and that G = {e, g, g2 , h, hg, hg2 }; then determine the possible multiplication tables of G.] 3. (a) Prove that x2 + x + 1 is irreducible in F2 [x], and use this to prove that there exists a field F4 with 4 elements; write out its multiplication table. (b) Similarly, show that x2 + 1 is irreducible in F3 [x], and that there is a field F9 with 9 elements; show that you multiply elements of F9 by the familiar rule (a + bi)(c + di) = ac − bd + (ad + bc)i. 4. (a) List all irreducible polynomials with coefficients in F2 of degree at most 4. (b) Use the previous results to explain how to construct a field with 8 elements and a field with 16 elements. 5. Prove that the polynomial ring k[x] over any field k has infinitely many irreducible polynomials. [Hint: Imitate Euclids proof that Z has infinitely many primes.] 6. (a) Show that x3 − 2x − 2 is irreducible in Q[x]. (b) Let a denote the image of x in Q[x]/(x3 − 2x − 2); write each of 1/a, 1/(1 + a) and 1/(1 + a2 ) in the form c2 a2 + c1 a + c0 with ci ∈ Q. 7. Show that x105 − 9 is irreducible in Q[x]. [Hint: we know all complex roots of this polynomial; use them to show that it is impossible to factorise this polynomial into a product of two polynomials of positive degrees with integer coefficients.] √ 8. √Find the polynomial with rational coefficients of smallest possible degree which has 2 + 3 as one of its roots.