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 \(a,b \in {\mathbb{F}}_q[x]\) 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 \(b\).