#8373: finite fields constructed with non-primitive defining polynomial
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Reporter: rkirov | Owner: cpernet
Type: enhancement | Status: new
Priority: minor | Milestone: sage-6.3
Component: finite rings | Resolution:
Keywords: | Merged in:
Authors: | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Description changed by jdemeyer:
Old description:
> Consider the following code:
> {{{
> sage: R.<x> = PolynomialRing(GF(2))
> sage: K.<a> = GF(16, modulus=x^4+x^3+x^2+x+1)
> sage: a^5
> 1
> }}}
>
> This is all fine mathematically, as long as the user is clear what a is
> and isn't (it isn't a generator for the multiplicative group of the
> finite field). So the options as I see them (in increasing difficulty for
> implementation):
>
> 1)GF already checks modulus for irreducibility, just add check for
> modulus.is_primitive().
>
> 2)Rewrite the help for the GF function to indicate that the function does
> not return a generator necessarily (like in this specific case).
>
> 3)Find an actual generator (that might not be the polynomial x) and
> return that.
>
> Opinions?
New description:
Add an argument `modulus="primitive"` to the finite field generator `GF()`
such that the chosen generator is guaranteed to be a generator of the
multiplicative group.
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Ticket URL: <http://trac.sagemath.org/ticket/8373#comment:5>
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