MATH3962: Rings fields and Galois Theory – Mathematics Assessment Answer

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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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