Polynomials over Finite Fields

The functions in this section are available for univariate polynomials over finite fields only.

PrimePolynomials(R, d) : RngUPol, RngIntElt -> SeqEnum[ RngUPolElt ]

PrimePolynomials(R, d, n) : RngUPol, RngIntElt, RngIntElt -> SeqEnum[ RngUPolElt ]

A sequence of all monic prime polynomials of R of degree d, resp. a sequence of n monic prime polynomials of R of degree d.

RandomPrimePolynomial(R, d) : RngUPol, RngIntElt -> RngUPolElt

A random monic prime polynomial of R of degree d.

NumberOfPrimePolynomials(q, d) : RngIntElt, RngIntElt -> RngIntElt

NumberOfPrimePolynomials(K, d) : FldFin, RngIntElt -> RngIntElt

NumberOfPrimePolynomials(R, d) : RngUPol, RngIntElt -> RngIntElt

The number of monic prime polynomials of degree d over the respective finite field.

JacobiSymbol(a,b) : RngUPol, RngUPol -> RngIntElt

The Jacobi symbol (a/b) of the two polynomials a, b ∈F_q[x] where q must be odd. If b is irreducible, the symbol equals 0 if b divides a. It equals 1 if a is a square mod b and -1 otherwise. The symbol then extends multiplicatively to all non-constant polynomials b.