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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 | --Copyright The Numerical Algorithms Group Limited 1994.
-- 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]
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