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Irreducible Polynomials

Find the polynomials that represent 1/x^3+x , x/x^3+x, x^2/x^3+x, and x^3/x^3+x modulo the irrreducible polynomial x^5+x^2+1 over F2 ( the field with two elements 0 and 1)

Your answers should be polynomials over F2 with degrees at most four. (Can you explain in here why at most degree four)

Note: Use the Eucliden algorithm for polynomials. (Can you do this algorithm step by step to understand)

Solution Preview

All the following calculation are in modulo x^5+x^2+1 and we note 1=-1 in F2.
By Euclidean algorithm, we have
0=x^5+x^2+1=(x^2-1)(x^3+x)+(x^2+x+1) => (x^2-1)(x^3+x)=x^2+x+1
x^3+x = ...

Solution Summary

Irreducible Polynomials are investigated.