Elements of function fields: extension#

class sage.rings.function_field.element_polymod.FunctionFieldElement_polymod#

Bases: FunctionFieldElement

Elements of a finite extension of a function field.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: x*y + 1/x^3
x*y + 1/x^3

element()#

Return the underlying polynomial that represents the element.

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<T> = K[]
sage: L.<y> = K.extension(T^2 - x*T + 4*x^3)
sage: f = y/x^2 + x/(x^2+1); f
1/x^2*y + x/(x^2 + 1)
sage: f.element()
1/x^2*y + x/(x^2 + 1)

is_nth_power(n)#

Return whether this element is an n-th power in the function field.

INPUT:

n – an integer

ALGORITHM:

If n is a power of the characteristic of the field and the constant base field is perfect, then this uses the algorithm described in Proposition 12 of [GiTr1996].

See also

nth_root()

EXAMPLES:

sage: # needs sage.rings.finite_rings
sage: K.<x> = FunctionField(GF(4))
sage: R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x)
sage: y.is_nth_power(2)
False
sage: L(x).is_nth_power(2)
True

list()#

Return the list of the coefficients representing the element.

If the function field is \(K[y]/(f(y))\), then return the coefficients of the reduced presentation of the element as a polynomial in \(K[y]\).

EXAMPLES:

sage: K.<x> = FunctionField(QQ); R.<y> = K[]
sage: L.<y> = K.extension(y^2 - x*y + 4*x^3)
sage: a = ~(2*y + 1/x); a
(-1/8*x^2/(x^5 + 1/8*x^2 + 1/16))*y + (1/8*x^3 + 1/16*x)/(x^5 + 1/8*x^2 + 1/16)
sage: a.list()
[(1/8*x^3 + 1/16*x)/(x^5 + 1/8*x^2 + 1/16), -1/8*x^2/(x^5 + 1/8*x^2 + 1/16)]
sage: (x*y).list()
[0, x]

nth_root(n)#

Return an n-th root of this element in the function field.

INPUT:

n – an integer

OUTPUT:

Returns an element a in the function field such that this element equals \(a^n\). Raises an error if no such element exists.

ALGORITHM:

If n is a power of the characteristic of the field and the constant base field is perfect, then this uses the algorithm described in Proposition 12 of [GiTr1996].