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PAPER TITLE: FURTHER PURE MATHEMATICS
EXAM DATE: FRIDAY 2, JUNE 2017
COURSE CODE: M303/B
Question 1
(a) (i) Use the Euclidean algorithm to find hcf(143,312).
(ii) Hence find integers s and t such that
143s + 312t = hcf(143,312).
(b) Prove by induction that
= for all positive integers n.
ANSWERS(Purchase full paper to get all the solutions):
1ai)
hcf(143,312) using Euclidean algorithm.
312 = 2(143) + 26
143 = 5(26) + 13
26 = 2(13) + 0
13 = 1(13) + 0
Therefore, the hcf = 13
1aii)
143s + 312t = hcf(143,312)
13 = 1(143) + (-5(26))
26 = (312 - 2(143))
13 = 1(143) + (-5((1(312) - 2(143))))
21 = 1(147) +10(143) + (-5(312))
13 = 11(143) + (-5(312))
11(143) + (-5(312)) = 13 (1)
143s + 312t = hcf(143,312) (2)
By comparing (1) and (2)
s = 11, and t = -5.
1b)
=
By mathematical induction.
For n = 1
L.H.S: 3n - 2 = 3(1) - 2 = 1
R.H.S:
L.H.S = R.H.S = 1, it is true
For n = 2
L.H.S: = 1+ (3(2) - 2) =5
L.H.S = R.H.S = 5, it is true
Assume, that n = k
= (1)
Show that n = k+1
= (2)
Add to both side of (1)
= (3)
Simplifying the right hand of (3)
The right-hand side of (3) = right-hand side of (2)
The left-hand side of (3) = the left-hand side of (2)
Therefore,
= →is true for all positive integer n.
Question 2
(a) Let G = Z4 × Z6.
(i) Write down the order of G.
(ii) Write down the order of (0,3).
(iii) Write down an element of order 6.
(iv) Is G cyclic? Justify your answer.
(b) Consider the dihedral group D12 given by
D12 ={r,s|r12 = s2 = e, sr = r11s}.
(i) Write the element r6sr6 in the form risj for some i ∈{ 0,...,11} and some j ∈{ 0,1}.
(ii) Find the order of the element r6s.
(iii) Write down a subgroup of D12 of order 12.
(iv) Is this subgroup normal? Justify your answer
Question 3
(a) Given 75 = 3×52,
(i) τ(75),
(ii) σ(75),
(iii) ∅(75).
(b) Calculate the following Legendre symbols (note that 179 and 199 are prime)
(i) (2/199),
(ii) (179/199).
(c) Let be a polynomial in Q[x].
(i) Show that .
(ii) Hence, or otherwise, prove that ) is irreducible in Q[x].
Question 4
Let
A = {(a,b) ∈ R2 :1≤ a2 + b2 ≤ 4},
B = {(a,b) ∈ R2 : |b| < 1} and
C = A∩B.
(a) Sketch on separate diagrams, or describe in words, each of A, B and C taking care to indicate which points lie in the sets.
(b) Write down the closure, interior and boundary of C for the Euclidean metric.
(c) Show that C is not connected for the Euclidean metric
(d) Determine the d0-closure of S = in R
Question 5
Let α = − ; you may assume that
[Q() : Q]=[Q() : Q] = 2, ∉ Q() and ∉ Q().
(a) Express α3 in the form a+b for suitable a,b ∈ Q.
(b) With the help of (a), or otherwise, show that ∈ Q(α).
(c) Hence show that Q(α) = Q().
(d) Determine the degree [Q(α):Q].
(e) By considering α2 find the minimal polynomial for α over Q
Question 6
Let A = [0 ,2) and B ={0}∪{2}.
(a) For each of the sets A and B, state whether or not they are d-connected, d-compact and d-complete when:
(i) d is the Euclidean metric for R.
(ii) d is the discrete metric for R.
(b) Give brief justifications of your answers for the sets A and B in the case when d is the discrete metric.
Question 7
Let G be a group of order 325 = 13×52.
(a) (i) Write down the order of the Sylow 5-subgroups of G.
(ii) Show that G has a unique Sylow 13-subgroup and a unique Sylow 5-subgroup.
(iii) Deduce that G is abelian.
(iv) Write down the non-isomorphic possibilities for G.
(b) Let G be a non-abelian group of order p3 where p is prime.
(i) Prove that the centre is isomorphic to Zp.
(ii) Deduce that G has a normal subgroup of order p2 (Hint: use the correspondence theorem.)
Question 8
Let R = Z[-7] = {a + b-7: a, b ∈ Z}, and let
N(a + b-) = a2 +7b2 be a norm on R, which you may assume is multiplicative.
(a) Show that the only elements r ∈ R with N(r) = 1 are r = ±1, and that there are no elements with N(r) = 2.
(b) Show that all of 2, 1+- are irreducible in R.
(c) Hence, or otherwise, show that R is not a UFD.
Let I ={2,1+} be the ideal in Z generated by the elements 2 and .
(d) Show that if α + β ∈ I for α,β ∈ Z, then α−β is even.
(e) By considering the cases where α,β ∈ Z are both even or both odd separately, prove that if α−β is even, then α + β ∈ I. Conclude that I = {α + β-; α,β ∈ Z,α−β is even}.
(f) With the help of (e) or otherwise, show that for every x ∈ Z [-;] such that x/ ∈ I we have 1 + x ∈ I.
(g) Using (f) or otherwise, show that I is a maximal ideal in Z [;].
QUESTION 9
Define a distance function d: Z × Z → R as follows:
(a) Prove that d is a metric on Z by showing that
(i) d satisfies (M1).
(ii) d satisfies (M2).
(iii) d satisfies (M3).
(b) Show that if a sequence (ai) in Z is eventually constant, then it is d-Cauchy.
(c) Show that if a sequence (ai) in Z is d-Cauchy, then it is eventually constant.
(d) Briefly explain whether (Z ,d) is a complete metric space.
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Last updated: Sep 02, 2021 03:02 PM
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