When defining a finite field of non-prime order, it is useful to give a name to the generator.
Likewise, when defining a polynomial ring, it is useful to give a name to the polynomial variable.
In the example, GF(2^8, 'a')
returns the finite field with $2^8$
elements, with a
as the display name of its generator.
And R.<x> = K[]
simultaneously defines R
as a polynomial ring
over the field K
, with a polynomial variable that displays as 'x',
and defines x
as its generator, i.e., the polynomial variable.
So x^254
is the monic monomial of degree 254 in this polynomial ring.
For more, read the documentation or/and the source code for GF
:
sage: GF?
sage: GF??
and for PolynomialRing
:
sage: PolynomialRing?
sage: PolynomialRing??
Note that R.<x> = K[]
is transformed by the Sage preparser into:
sage: preparse("R.<x> = K[]")
"R = K['x']; (x,) = R._first_ngens(1)"
Note: this is an excerpt of the