Numbers and Sets | Part IA, 2008

(a) Let F\mathbb{F} be a field such that the equation x2=1x^{2}=-1 has no solution in F\mathbb{F}. Prove that if xx and yy are elements of F\mathbb{F} such that x2+y2=0x^{2}+y^{2}=0, then both xx and yy must equal 0 .

Prove that F2\mathbb{F}^{2} can be made into a field, with operations

(x,y)+(z,w)=(x+z,y+w)(x, y)+(z, w)=(x+z, y+w)


(x,y)(z,w)=(xzyw,xw+yz).(x, y) \cdot(z, w)=(x z-y w, x w+y z) .

(b) Let pp be a prime of the form 4m+34 m+3. Prove that 1-1 is not a square (modp)(\bmod p), and deduce that there exists a field with exactly p2p^{2} elements.

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