Explore BrainMass
Share

Field and ring proofs

This content was STOLEN from BrainMass.com - View the original, and get the already-completed solution here!

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)

© BrainMass Inc. brainmass.com October 25, 2018, 12:10 am ad1c9bdddf
https://brainmass.com/math/ring-theory/field-ring-proofs-218847

Solution Summary

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

$2.19
See Also This Related BrainMass Solution

Ring theory proof fields

Modern Algebra
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.

View Full Posting Details