# ConwayPolynomials -- database of Conway polynomials for use with GF

## Description

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.

The package is loaded by default when Macaulay2 starts up. The database is loaded the first time conwayPolynomial is called, resulting in a brief pause.

In this example, we show how GF and map behave when the package is present.

 i1 : GF 125 o1 = GF 125 o1 : GaloisField i2 : ambient oo ZZ --[a] 5 o2 = ----------- 3 a - 2a - 2 o2 : QuotientRing i3 : map(GF 125^2, GF 125) 5 4 3 2 o3 = map (GF 15625, GF 125, {- a + a - a - a - 2a - 2}) o3 : RingMap GF 15625 <--- GF 125 i4 : isWellDefined oo o4 = true

## Author

• Daniel R. Grayson

## Version

This documentation describes version 1.0 of ConwayPolynomials.

## Source code

The source code from which this documentation is derived is in the file ConwayPolynomials.m2.

## For the programmer

The object ConwayPolynomials is .