Exercises 1

1. Use the axioms for a ring to prove that 0 . a = 0 for any element a of the ring.
   Prove that -1 . -1 = 1 and thereby justify the old rhyme:

   Minus times minus is equal to plus
   The reasons for this we will not discuss.

   Solution to question 1

2. Let S be any set and let T = Map(S, R) the set of maps from S to R. Define + and . to be the usual (pointwise) addition and multiplication of functions:
   
   (f + g)(x) = f(x) + g(x) and (f . g)(x) = f(x).g(x) for all x ∈ S.
   
   Prove that T is a ring under these operations. What is the multiplicative identity of T? Which elements of T have multiplicative inverses?
   If S = {a, b} is a set with just two elements, show that the set T can be put into one-one correspondence with the set R². What happens to the ring operations under this correspondence?

   Solution to question 2

   Prove that the set { a + b√2 | a, b ∈ Z} is a ring under real addition and multiplication. Find the multiplicative inverses of the elements 1 + √2 and 3 + 2√2.
   Can you characterise those elements which have multiplicative inverses?

   Solution to question 3

   If ω is a complex cube root of 1, prove that the subset R = {a + bω + cω²| a, b, c ∈ Z} of C is a ring under the usual addition and multiplication of complex numbers.
   Draw the positions of elements of R in the Argand diagram.
   Calculate which elements of R have multiplicative inverses which are in R and indicate them on your diagram.
   [Hint: if |z| > 1 for a complex number z then |z^-1| <1]

   Solution to question 4

   Let U = Map(R, R) with the usual addition of functions and composition of functions for the multiplicative operation. What is the multiplicative identity? Which elements of U have multiplicative inverses?
   Prove that (f + g)h = fh + gh for all f, g, h ∈ U but that in general f(g + h) ≠ fg + fh and so U is not a ring under these operations.

   Solution to question 5

   The section of Diophantus's Arithmetica that Fermat was reading when he wrote his famous marginal note was about Pythagorean triples.
   A Pythagorean triple (a, b, c) of positive integers satisfies a² + b² = c².
   a)  Show that if we can find positive integers p and q such that a = p² - q², b = 2pq, c = p² + q² then (a, b, c) is a Pythagorean triple.
   b)  If (a, b, c) is a Pythagorean triple and a, b are coprime (such a Pythagorean triple is called primitive) prove that exactly one of a, b is even. [Hint: work modulo 4]
   c)  If b is the even one prove that one can find positive integers p and q such that a = p² - q², b = 2pq and c = p² + q².
   d)  Hence list all the primitive Pythagorean triples (a, b, c) with c < 60.
   e)  Find a Pythagorean triple with the ratio a/b as close to 1 as you can. In other words, find an integer-sided right-angled triangle as close as you can to isosceles.

   Solution to question 6