ApCoCoALib:RingF16

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ApCoCoA will soon contain an implementation of the field $\mathbb {F} _{16}$ The field is constructed via $\mathbb {F} _{16}=\mathbb {F} [x]/(x^{4}+x^{3}+1)$. The field's elements are represented as integers between 0 and 15. The corresponding mapping is the substitution homomorphism, mapping x to 2. Therefore we have e.g. $x^{3}+x+1\mapsto 2^{3}+2^{1}+2^{0}=8+2+1=11$

alternative representations

Instead of using $x^{4}+x^{3}+1$, we could have also chosen another irreducible polynomial of degree 4. In total, there are three ireducible ones, namely $\{x^{4}+x^{3}+1,x^{4}+x^{3}+1,x^{4}+x^{3}+x^{2}+x+1\}$ If you have a system, which is based on one of the other irreducibly polynomials, you have to construct an isomorphism between the different representations of the field (which is unique). These isomorphisms can be constructed by mapping the irreducible polynomial's roots to the roots of x^4 + x^3 + 1, respecting the fields galois-group.

The irreducible polynomials and roots are:

[x^4 + x^3 + 1, [x^2, x, x^3 + x^2 + x, x^3 + 1]]
[x^4 + x + 1, [x^2, x, x^2 + 1, x + 1]]
[x^4 + x^3 + x^2 + x + 1, [x^3, x^2, x, x^3 + x^2 + x + 1]]

They were produced with the CoCoA4 code, explained here.