CoCoA » CoCoALib

• IsPthPower
• IsPthPower2 says whether arg is a P^k-th power (where k is a given arg)
• PthRoot computes a P-th root
• PthRoot2 computes a P^k-th root

The names of these fns are open to discussion!

PS the PthRoot can be computed using linear algebra!

First impls have been made.

The cardinality is represented by a fn called CardExp which gives the exponent of p needed to obtain the cardinality -- this seems to be most useful quantity (and it is trivial to obtain the actual cardinality).

• prime finite field
• zero-dim quotient of polyring over a finite field

A "convoluted" construction which happens to produce a finite field will be declared not a finite field. For example if P is ZZ[x] then the quotient P/I will be declared not a finite field regardless of the value of I.

I really don't like the name CardExp -- I think it is too cryptic.

The value it returns is log(cardinality)/log(characteristic), so in some ways LogCard or LogCardinality are more meaningful; though I am slightly unhappy that these names do not start Card....

Right now I'd say that LogCardinality is easily the clearest candidate.

What should it do if its arg is not a finite field? Error? Return 0?
Perhaps giving an error is best because (in CoCoALib) it is not really defined for any other sort of ring.

Maybe some clever person in the future will be able to extend the fn to work for algebraic extensions of finite non-fields should there ever be a need.

Where should the documentation go?

JAA now believes that he has a decent algorithm for testing a polynomial ideal for maximality. As prerequisite it needs a function to convert a poly into a vector of coeffs WRT to a given PP basis. It also needs univariate poly factorization, but we already have that.

Added C5 documentation for IsFiniteField, LogCardinality, IsPthPower and PthRoot.

JAA wonders whether ExtnDeg may not be a better name for LogCardinality. The name is clearer and can be applied more widely (e.g. to algebraic extensions over QQ).

Should there also be a fn called PthPower?

I note that IsPthPower2 and PthRoot2 have become forgotten; perhaps they should be put on hold until a real need for them arises. If we make them then we should also consider PthPower2. Alternative names could be QthRoot except that "q" tends to mean the cardinality of the finite field rather than any power of "p".

Implementation of IsMaximal for multivariate rings now allows working non-simple extensions.