\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/input ffx72.input}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{License}
<<license>>=
--Copyright The Numerical Algorithms Group Limited 1994.
@
<<*>>=
<<license>>

-- This file demonstrates some calculations with the finite field of
-- 49 elements. It is built as a degree 2 extension of the Galois
-- field with 7 elements.
)cl all
)time off

gf72 := FF(7, 2)
-- x^2+1 is irreducible over PF 7

u: UP(x,PF 7) := x^2 + 1
factor u

-- but factors over FF(PF 7, 2)

u2 : UP(x,gf72) := u
factor u2

-- the following is the irreducible polynomial used in the representation
-- of GF(7^2) over PF 7. It will be the same every time this field is
-- used.

definingPolynomial()$gf72

-- e is a randomly chosen element

e := index(size()$gf72 quo 3)$gf72
norm e
trace e

-- the order of an element is the minimum positive integer to which
-- it can be raised to yield 1.

order e

-- we can display all the nonzero elements in the field

allElts := [index(i :: PI)$gf72 for i in 1..48]

-- we can sum over them

reduce(+,allElts)

-- and we can determine the order of each of them.  Each element of
-- order 48 generates the multiplicative group of non-zero elements.

[order e for e in allElts]
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
