MATH20212 Algebraic Structures 2
Coursework Take Away Test
1. Let Show that S is a subring of M2
Is S a domain? Explain your answer. [6 marks]
2. Write down all the zero divisors in Z8 .
Find a degree 2 polynomial in Z8 [X] with exactly 4 roots in Z8 and list the roots of this polynomial. [4 marks]
3. Consider the map θ : Z → Z6 × Z4 , defined by θ(a) = ([a]6, [a]4 ) for all a ∈ Z. Show that θ is a ring homomorphism.
Describe the kernel of θ and find a generator for ker(θ) as a principal ideal. [5 marks]
4. Let I = {a + b√2 | a, b ∈ Z, a is even}.
Show that I is an ideal of Z[√2] = {a + b√2 | a, b ∈ Z}.
Explain why I contains no units. [5 marks]