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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 | )abbrev category FPC FieldOfPrimeCharacteristic
++ Author: J. Grabmeier, A. Scheerhorn
++ Date Created: 10 March 1991
++ Date Last Updated: 31 March 1991
++ References:
++ Grab92 Finite Fields in Axiom
++ Description:
++ FieldOfPrimeCharacteristic is the category of fields of prime
++ characteristic, for example, finite fields, algebraic closures of
++ fields of prime characteristic, transcendental extensions of
++ of fields of prime characteristic.
FieldOfPrimeCharacteristic() : Category == SIG where
SIG ==> Join(Field,CharacteristicNonZero) with
order : $ -> OnePointCompletion PositiveInteger
++ order(a) computes the order of an element in the multiplicative
++ group of the field.
++ Error: if \spad{a} is 0.
discreteLog : ($,$) -> Union(NonNegativeInteger,"failed")
++ discreteLog(b,a) computes s with \spad{b**s = a} if such an s exists.
primeFrobenius : $ -> $
++ primeFrobenius(a) returns \spad{a**p} where p is the characteristic.
primeFrobenius : ($,NonNegativeInteger) -> $
++ primeFrobenius(a,s) returns \spad{a**(p**s)} where p
++ is the characteristic.
add
primeFrobenius(a) == a ** characteristic()
primeFrobenius(a,s) == a ** (characteristic()**s)
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