ConwayPolynomials is a package that provides a database of Conway polynomials. A Conway polynomial for a prime p and an exponent n is a particular monic polynomial whose roots are primitive elements of a finite field with p^n elements. They can take a very long time to compute, so the package comes with just a publicly available database of Conway polynomials provided by Frank Luebeck.
After the package is loaded, the function GF will return Galois fields presented by Conway polynomials, provided they are in the table. Moreover, map can be used to produce the canonical maps between Conway Galois fields.
i1 : conwayPolynomial 125 3 o1 = a - 2a - 2 ZZ o1 : --[a] 5 |
i2 : GF 125 o2 = GF 125 o2 : GaloisField |
i3 : ambient oo ZZ --[a] 5 o3 = ----------- 3 a - 2a - 2 o3 : QuotientRing |
i4 : map(GF 125^2, GF 125) 5 4 3 2 o4 = map(GF 15625,GF 125,{- a + a - a - a - 2a - 2}) o4 : RingMap GF 15625 <--- GF 125 |
i5 : isWellDefined oo o5 = true |