Define the ring of quaternions.
Section A
1. Suppose that V is a finite-dimensional vector space over a field K and that α ∈ End(V ).
(a) If λ is an eigenvalue of α, define what is meant by the algebraic multiplicity of λ
and the geometric multiplicity of λ.
(b) Take K = F13, the field with 13 elements, and V = F
3
13. Consider the 3 × 3 matrix
with entries in F13
A =
6 2 0
10 5 3
8 2 7
and the corresponding element of End(V ). If
x =
1
3
8
and y =
3
1
−1
are elements of F
3
13, compute (A + I)x and (A − 3I)
2y.
(c) Write down the JNF of A, explaining why your answer is correct.
[Question 1 carries 8 marks out of 75]
2. (a) Define the ring of quaternions, H.
(b) Use quaternions to write 33952541 as a sum of four squares. You are allowed to use
the fact that 33952541 = 8771 × 3871, and you may find it helpful to compute 622
and 932
.
[Question 2 carries 8 marks out of 75]
Page 2 of 5 MA20217
3. In this question, R and S are arbitrary rings.
(a) Define what it means for a map ϕ: R → S to be a ring homomorphism. Give an
example to show that the definition does not imply that ϕ(1R) = 1S.
(b) Define what it means for a map ϕ: R → S to be an isomorphism of rings. Show
that a bijective ring homomorphism is an isomorphism of rings.
(c) Show that if ϕ: R → S is a ring homomorphism and I is an ideal contained in Ker ϕ,
then there is a unique ring homomorphism ¯ϕ: R/ Ker ϕ → S such that ϕ = ¯ϕπ,
where π : R → R/I is the quotient map.
[Question 3 carries 8 marks out of 75]
Section B
4. (a) Define what it means for a ring R to be an integral domain.
(b) Let R be a ring with 0R 6= 1R. Show that R is an integral domain if and only if the
following holds: if a, b, c ∈ R and a 6= 0R, and ab = ac, then b = c.
(c) Show that if R is an integral domain then the characteristic of R is either zero or a
prime.
(d) Briefly explain what is meant by the field of fractions Q(R) of an integral domain R
(you do not need to give the full definition).
(e) Suppose that R is an integral domain, F = Q(R) and S is a proper subring of F
with 1R ∈ S, such that Q(S) = F. Does this necessarily imply that S = R? You
must give a proof or a counterexample.
(f) Suppose that R is an integral domain. For each of the following rings A, say whether
A is always a domain; never a domain; or possibly a domain, depending on what R
is. Give brief proofs or counterexamples.
(i) A is the direct product R × R.
(ii) A is the ring of formal power series R[[t]].
(iii) R/6R, where 6 means 1R + 1R + 1R + 1R + 1R + 1R.
[Question 4 carries 18 marks out of 75]
Page 3 of 5 MA20217
5. In this question R is always an integral domain.
(a) Define what is meant by a valuation on R, and what is meant by a Euclidean
valuation.
(b) Define what it means for R to be a principal ideal domain (also known as a PID),
and what it means for R to be a Euclidean domain.
(c) Show that if R is a Euclidean domain then R is a PID.
(d) Suppose that R is a PID and S is a subring of R containing 1R. Is S necessarily a
PID? Give a proof or counterexample.
(e) Suppose that R is a PID and I is an ideal of R, with I 6= R. Is the quotient ring
R/I necessarily a PID? Give a proof or counterexample.
(f) By considering the ring R[s, t], or otherwise, give an example of a valuation that is
not a Euclidean valuation.
[Question 5 carries 18 marks out of 75]
6. In this question, R is always a UFD and F = Q(R) is its field of fractions.
(a) Explain briefly what is meant by the content of a polynomial f ∈ R[t].
(b) Define what it means for a polynomial f ∈ R[t] to be primitive.
(c) Suppose that f ∈ R[t] is primitive, and that f is irreducible in F[t]. Show that f is
irreducible in R[t].
(d) In each of the following cases, A is a UFD and f ∈ A: say whether f is irreducible
in A. If it is not, give a factorisation. Justify your answers.
(i) A = Z[i] and f = 5.
(ii) A = Z[i] and f = 7.
(iii) A = Z[t] and f = 6t
2 + 3t + 9.
(iv) A = R[s, t] and f = s
2 + t
2
.
(v) A = C[s, t] and f = s
2 + t
2
.
(vi) A = C[s, t, u] and f = s
2 + t
2 + u
2
.
[Question 6 carries 18 marks out of 75]
Page 4 of 5 MA20217
Section C
7. Write a description, in your own words, of ONE of the following topics.
You should explain some, but not all, of the main results and the main steps towards
proving them, but not give full detailed proofs.
Abbreviations (such as UFD and JNF, for example) that were used in the lectures may
be used without explanation.
The word limit is approximately 500 words.
(a) Unique factorisation.
(b) Jordan normal form.
[Question 7 carries 15 marks out of 75
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