MATH3907 ⋅ SESSION 1, 2022 ASSIGNMENT 2 ⋅ SOLUTIONS
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MATH3907 ⋅ SESSION 1, 2022
ASSIGNMENT 2 ⋅ SOLUTIONS
1. [8 marks]. Find all solutions to the following systems of congruences.
(
) 
≡6 2 
≡4 3
(
) 
≡5 4
≡7 1
≡11 3
(
) 
≡15 11
≡20 16
2. [4 marks]. Find all solutions to the following system of Diophantine equations
2
+ 15
= 7
3
+ 20
= 8.
3. [4 marks]. Let 
be a ring and 
, 
∈ 
. Show that
(a) if 
+ 
= 0 then 
+ 
= 0
(b) if 
+ 
= 0 and 
is commutative then (
+ 
)2 = 
2 + 
2.
4. [8 marks]. In group theory, you met the six-element abelian group
ℤ2 × ℤ3 = {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2)}
with group operation given by componentwise addition (mod 2 in the first component and mod 3 in the second component). In this question you are going to investigate ways in which this could be equipped with a multiplication making it into a ring.
(a) Using the fact that (1, 0) + (1, 0) = (0, 0), show that (1, 0)(1, 0) is either (1, 0) or (0, 0). (Hint: you could use the previous question.)
(b) What does the fact that (0, 1)+(0, 1)+(0, 1) = (0, 0) tell you about the possible values of (0, 1)(0, 1)?
(c) What are the possible values of (1, 0)(0, 1)?
(d) Does there exist a field with 6 elements?
5. [6 marks]. Determine whether or not each of the following polynomials is irreducible over the integers.
(a) [2 marks]. 
4 − 4
− 8
(b) [2 marks]. 
4 − 2
− 6
(c) [2 marks]. 
4 − 4
2 − 4
6.
[7 marks]. This question is about the ring ℤ11 of integers mod 11.
(a) [1 marks]. Is ℤ11 a field?
(b) [2 marks]. For which values of 
∈ ℤ11 does the equation 
2 + 
= 
have a solution?
(c) [2 marks]. For which values of 
∈ ℤ11 is the ring ℤ11 [
]/(
2 + 
+ 
) a field?
(d) [2 marks]. Explain why there is no homomorphism ℤ11 [
]/(
2 + 
+ 1) → ℤ11.
2022-07-09