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Field and ring proofs

1. Define F sub four to be the set of all 2x2 matrices.

F(sub 4)= [ a b ] ; a,b elements of F sub 2
b a+b

i) Prove that F sub four is a commutative ring whose operations are matrix addition and matrix multiplication
ii) prove that F sub four is a field having exactly four elements
iii) show that I sub four is not a field

F sub four= finite field having 4 elements
F sub two= finite field having 2 elements
I sub four= integers modulo four

2. Show that if f(x)= (x^p)-x element F sub p[x], then its polynomial function (f^beta): F sub p ---->F sub p is identically zero

(capital F sub p is finite field having p elements.
f^beta is function raised to power of beta)

Solution Summary

This provides several examples of working with proofs regarding fields and rings.