MATH3962: Rings fields and Galois Theory - Mathematics Assessment Answer

Subject Code: MATH3962

Mathematics Assessment Answer

Task: 1. (a) Provide an example for each of the following. In each case give a proof that your example satisfies the given conditions. (i) An irreducible element in the ring Q[x]. (ii) A maximal ideal in Z[x]. (iii) A unit u 6 = 1 in the ring Z[x]/(x 2 + 3x + 1)Z[x]. ? (iv) An irreducible element in Z[ ?3] that is not prime. (b) Write 45 + 420i as a product of irreducible Gaussian integers, showing all working. (c) Let R, S, T be rings, and suppose that ? : R ? S and ? : R ? T are ring homomorphisms. Show that the map ? : R ? S × T with ?(x) = (?(x), ?(x)) is a ring homomorphism. (d) Show that if n = p a q b with p, q > 1 distinct primes and a, b ? 1 integers then Z/nZ ? = (Z/p a Z) × (Z/q b Z). 2. (a) Let a(x), b(x) ? Q[x] be the polynomials a(x) = x 6 ? 2x 5 ? x 4 + 5x 3 ? 2x 2 ? 2x + 2 b(x) = x 5 ? 3x 4 + 3x 3 ? 2x + 2. Find a generator of the principal ideal a(x)Q[x] + b(x)Q[x], showing all working. (b) Prove or disprove: (i) If F is a field, and R is a nontrivial ring, and ? : F ? R is a nontrivial ring homomorphism, then ? is injective. (ii) The set of real numbers R equipped with addition ? and multiplication a ? b = min{a, b} and a defined by b = a + b for a, b ? R (here “+” is the usual addition on R) is a ring. (iii) There exists an ideal I of Z 2 [x] such that Z 2 [x]/I ? = Z 2 × Z 2 . (c) You are given that up to isomorphism there are exactly 4 distinct unital rings with precisely 4 elements. Find them all. 3. (a) Find all ideals J of Z[x] with xZ[x] ? J ? Z[x]. ? (b) Let R = {a + b ?11 | a, b ? Z or a, b ? Z + 2 1 }. Here Z + 12 denotes the set of all numbers of the form n + 12 with n ? Z (the half-integers). You are given that R is a commutative unital subring of C (you do not need to prove this). (i) Let N : R ? [0, ?) be given by N (z) = |z| 2 . Show that N (z) ? N for all z ? R. (ii) Find all units of R. ? (iii) Decompose 25 ? 2 1 ?11 into irreducible factors over R. 2 (iv) Show that R is a principal ideal domain. The University of Sydney - School of Mathematics and Statistics

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