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
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)
This provides several examples of working with proofs regarding fields and rings.
Ring theory proof fields
Ring Theory (IX)
The Field of Quotients of an Integral Domain
Prove that the mapping φ:D→F defined by φ(a) = [a , 1] is an isomorphism of D into F ,
where D is the ring of integers and F is the field of quotients of D.