[GAP Forum] How do I study fields like ZN[X}/<poly> in GAP
Sven Reichard
Sven.Reichard at tu-dresden.de
Fri Jun 28 13:59:41 BST 2013
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Am 27.06.2013 23:52, schrieb Sopsku:
> Dear Forum,
>
> I am new to GAP and trying hard to come up to speed with all the
> GAP documentation.
>
> In particular I can't figure out which functions are available to
> set up and solve the following (rather simple) problem:
>
> Consider the a finite field over n elements over a polynomial, for
> example, the field Z_2[X]/<x^2+x+1>. Find the elements and give the
> addition and multiplication tables.
>
> (I have no problem solving this "by hand" (or using Mathematica),
> but I can't seem to figure how to get the job done in GAP)
>
> Thank you for any specific solution or even pointers to the GAP
> functions that I should be using. Sopsku
>
Dear Sopsku,
here's one possibility. Probably there are more elegant ways.
gap> x := Indeterminate(GF(2), "x");
x
gap> r := PolynomialRing(GF(2));
GF(2)[x]
gap> p := x^2+x+1;
x^2+x+Z(2)^0
gap> i := Ideal(r, [p]);
<two-sided ideal in GF(2)[x], (1 generators)>
gap> quotient := r/i;
<ring GF(2),(1),(x)>
gap> Size(last);
4
gap> # addition table
gap> PrintArray(List(quotient, x -> List(quotient, y -> x+y)));
[ [ 0*(1), (x), (1), (1)+(x) ],
[ (x), 0*(1), (1)+(x), (1) ],
[ (1), (1)+(x), 0*(1), (x) ],
[ (1)+(x), (1), (x), 0*(1) ] ]
gap> # multiplication table
gap> PrintArray(List(quotient, x -> List(quotient, y -> x*y)));
[ [ 0*(1), 0*(1), 0*(1), 0*(1) ],
[ 0*(1), (1)+(x), (x), (1) ],
[ 0*(1), (x), (1), (1)+(x) ],
[ 0*(1), (1), (1)+(x), (x) ] ]
Cheers,
Sven Reichard.
- --
Institut für Algebra
TU Dresden
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