Missing (?) or undocumented (?) functionality for finite fields:

[fingolfin]

Given a finite field in Nemo, how can I...

• obtain a primitive root / multiplicative generator of (the multiplicative group of) the field?
• compute the multiplicative order of a non-zero element?
• get a basis over the prime field (actually, this one is there under the name basis -- it is merely missing in the manual)
• get a basis over some subfield (?)
• decompose an element in terms of that basis (or some other other basis even?)

[thofma]

The multiplicative functionality is not in Nemo, so the answer to the first two points is "you can't". It is in Hecke though.

• get a basis over some subfield (?)

What does subfield mean in this context? Given by an emedding? If L/K was constructed as an extension of K, than basis(L) will be the basis with respect to the base field K.

• decompose an element in terms of that basis (or some other other basis even?)

There is coeff(a, n), so that a == sum(coeff(a, i) * b for (i, b) in zip(degree(K), basis(K)). Not a big fan of this name though. I would prefer coordinates for the whole list and coordinate.

Edit:

• get a basis over the prime field (actually, this one is there under the name basis -- it is merely missing in the manual)

Should be absolute_basis.

[fingolfin]

Indeed, there is Hecke.primitive_root, thanks.