Rings and Fields

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(Exercises 2)

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 xS.
    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

  3. Prove that the set { a + b√2 | a, bZ} 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

  4. If ω is a complex cube root of 1, prove that the subset R = {a + bω + cω2| a, b, cZ} 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

  5. 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, hU but that in general f(g + h) ≠ fg + fh and so U is not a ring under these operations.

    Solution to question 5

  6. 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

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(Exercises 2)

JOC/EFR 2004