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LANCASTER UNIVERSITY
2011 EXAMINATIONS
PART II (Third or Fourth Year)
MATHEMATICS & STATISTICS 2 hours
Math 322: Rings, Fields and Polynomials
You should answer ALL Section A questions and TWO Section B questions.
In Section A there are questions worth a total of 50 marks, but the maximum mark that you can
gain there is capped at 40.
SECTION A
A1. Showing details of your working, find all the invertible elements of the ring Z 24. [6]
A2. Let p, q, r and s be pairwise non-associated, irreducible elements of a unique factorization
domain R. Find a highest common factor of the three elements
a = p
3 q
4 s
5 , b = p
2 r
2 s
6 and c = p
4 qr
3 s
2
in R. No justification of your answer is required. [3]
A3. Find the remainder on dividing the polynomial f (X) = ̂ 3 X^5 + ̂ 2 X^4 + X^2 + ̂ 2 X + ̂3 by X + ̂ 3
in Z 5 [X]. [4]
A4. Explain why each of the following three polynomials is irreducible as an element of the
specified polynomial ring. You may use general results without proof provided that you give
clear reference to them.
(a) g(X) = − 90 X^7 + 130X^4 − 26 X^3 + 156X + 78 in Q[X] [6]
(b) h(X) = 2X^2 − 6 X + 5 in R[X] [2]
(c) k(X) = (1 − 3
2 i)X + 2 + 7
2 i in (Z[
2 i])[X] [7]
please turn over
SECTION A continued
A5. Let
(X) =
X
5
X
4
X
2 −
X −
∈ Q[X].
(a) Write (X) as the product of a rational number and a primitive polynomial over Z. [6]
(b) Find all possible ways of writing (X) as the product of a rational number and a primitive
polynomial over Z. No justification of your answer is required. [2]
A6. (a) State Gauss’s Factorization Theorem. [5]
(b) Let f (X) be a primitive polynomial over a unique factorization domain R, and let F
be the field of fractions of R. Use Gauss’s Factorization Theorem to help prove that if
f (X) is irreducible in R[X], then f (X) is also irreducible in F [X]. [9]
please turn over
SECTION B continued
B2. (a) Let^ R^ be a commutative ring with a multiplicative identity.^ Define what is meant by
saying that an element a of R is irreducible. [2]
(b) Suppose that R is a principal ideal domain. Prove that if a is an irreducible element of
R, then the ideal aR generated by a is a maximal ideal in R. (You do not need to verify
that aR is an ideal in R.) [8]
(c) Explain briefly why Z 7 [X] is a principal ideal domain. [2]
(d) Show that the polynomial
p(X) = X^3 + ̂ 2 X^2 + ̂ 3 X + ̂ 5
is irreducible in Z 7 [X]. [6]
(e) Let I be the ideal in Z 7 [X] generated by the polynomial p(X) defined in (d).
(i) Explain briefly why the quotient ring Z 7 [X]/I is a field, and determine the number
of elements of this field, giving clear reference to any general results that you use. [3]
(ii) Let f (X) = ̂ 2 X^2 + ̂ 3 X + ̂ 5 ∈ Z 7 [X], and let π : Z 7 [X] → Z 7 [X]/I be the quotient
homomorphism. Explain why the element π
f (X)
is invertible in Z 7 [X]/I, and
find its multiplicative inverse. [9]
B3. (a) Let
Q[
3] =
q + r
3 : q, r ∈ Q
and Z[
3] =
m + n
3 : m, n ∈ Z
You may assume without proof that Z[
3] is a subring of R.
(i) Show that Q[
3] is a subfield of R. [5]
(ii) Show that Q[
3] is the field of fractions of Z[
3]. [3]
(b) Let F be a finite field.
(i) Show that F contains a smallest subfield F 0 (the prime subfield). [6]
(ii) Let ξ : Z → F 0 be the unital ring homomorphism given by ξ(m) = m · (^1) F for each
m ∈ Z, where 1F denotes the multiplicative identity of F. (You do not need to
verify that ξ is a unital ring homomorphism.) Find the kernel of ξ, and show that
the field F 0 from (i) is isomorphic to Zp for some prime number p. You may use
general results without proof provided that you give clear reference to them. [11]
(iii) Prove that F has pn^ elements for some prime number p and some n ∈ N. [5]
end of exam
