Previous page (Contents) | Contents | Next page (Exercises 2) |

- 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. - 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**R**^{2}. What happens to the ring operations under this correspondence?- 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?- If
*ω*is a complex cube root of 1, prove that the subset*R*= {*a*+*b**ω*+*c**ω*^{2}|*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]- 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*=*f**h*+*g**h*for all*f*,*g*,*h*∈*U*but that in general*f*(*g*+*h*) ≠*f**g*+*f**h*and so*U*is*not*a ring under these operations.- 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*a*^{2}+*b*^{2}=*c*^{2}.

a) Show that if we can find positive integers*p*and*q*such that*a*=*p*^{2}-*q*^{2},*b*= 2*pq*,*c*=*p*^{2}+*q*^{2}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*=*p*^{2}-*q*^{2},*b*= 2*pq*and*c*=*p*^{2}+*q*^{2}.

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. - (

Previous page (Contents) | Contents | Next page (Exercises 2) |