Friday, 10 July 2015

Abstract algebra, Field extension

Suppose F and H are fields of size q=prcontainingGF(p)as subfield.αis a primitive element of F and β is a primitive element of H.m(x)is the minimal polynomial of α.Non-zero Elements of both fields satisfy the equationxq1=1.m(x)is divisor of xq11.Hence there is an element in H (say)βtwhich is a root m(x).
I want to show that there exists a field isomorphism ϕ:FH which carries zero to zero and αto βt

Can you help to prove it? I have tried ϕ(αj)=βtj but I could not show that ϕ preserves addition.( This argument is presented by Vera Pless in the book Introduction to Theory of Error correcting codes.)

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