# Polynomials¶

This section contains classes and functions for creating polynomials over Galois fields.

## Polynomial classes¶

 Poly(coeffs[, field, order]) A univariate polynomial $$f(x)$$ over $$\mathrm{GF}(p^m)$$.

## Special polynomials¶

### Irreducible polynomials¶

 irreducible_poly(order, degree[, method]) Returns a monic irreducible polynomial $$f(x)$$ over $$\mathrm{GF}(q)$$ with degree $$m$$. irreducible_polys(order, degree[, reverse]) Iterates through all monic irreducible polynomials $$f(x)$$ over $$\mathrm{GF}(q)$$ with degree $$m$$.

### Primitive polynomials¶

 primitive_poly(order, degree[, method]) Returns a monic primitive polynomial $$f(x)$$ over $$\mathrm{GF}(q)$$ with degree $$m$$. primitive_polys(order, degree[, reverse]) Iterates through all monic primitive polynomials $$f(x)$$ over $$\mathrm{GF}(q)$$ with degree $$m$$. conway_poly(characteristic, degree) Returns the Conway polynomial $$C_{p,m}(x)$$ over $$\mathrm{GF}(p)$$ with degree $$m$$. matlab_primitive_poly(characteristic, degree) Returns Matlab's default primitive polynomial $$f(x)$$ over $$\mathrm{GF}(p)$$ with degree $$m$$.

### Interpolating polynomials¶

 lagrange_poly(x, y) Computes the Lagrange interpolating polynomial $$L(x)$$ such that $$L(x_i) = y_i$$.

## Polynomial functions¶

### Divisibility¶

 Finds the greatest common divisor of $$a$$ and $$b$$. Finds the multiplicands of $$a$$ and $$b$$ such that $$a s + b t = \mathrm{gcd}(a, b)$$. Computes the least common multiple of the arguments. Computes the product of the arguments. Determines if the arguments are pairwise coprime.

### Congruences¶

 Solves the simultaneous system of congruences for $$x$$.

### Factorization¶

 Computes the prime factors of a positive integer or the irreducible factors of a non-constant, monic polynomial.

### Tests¶

 Determines if an integer or polynomial is square-free.

Last update: May 18, 2022