# Galois Fields¶

This section contains classes and functions for creating Galois field arrays.

## Galois field classes¶

### Class factory functions¶

 GF(order[, irreducible_poly, ...]) Creates a FieldArray subclass for $$\mathrm{GF}(p^m)$$. Field(order[, irreducible_poly, ...]) Alias of GF().

### Abstract base classes¶

 FieldArray(x[, dtype, copy, order, ndmin]) A ndarray subclass over $$\mathrm{GF}(p^m)$$.

### Pre-made FieldArray subclasses¶

 GF2(x[, dtype, copy, order, ndmin]) A ndarray subclass over $$\mathrm{GF}(2)$$.

## Prime field functions¶

### Primitive roots¶

 primitive_root(n[, start, stop, method]) Finds a primitive root modulo $$n$$ in the range [start, stop). primitive_roots(n[, start, stop, reverse]) Iterates through all primitive roots modulo $$n$$ in the range [start, stop). Determines if $$g$$ is a primitive root modulo $$n$$.

## Extension field functions¶

### 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 $$m$$.

### Primitive elements¶

 primitive_element(irreducible_poly[, method]) Finds a primitive element $$g$$ of the Galois field $$\mathrm{GF}(q^m)$$ with degree-$$m$$ irreducible polynomial $$f(x)$$ over $$\mathrm{GF}(q)$$. primitive_elements(irreducible_poly) Finds all primitive elements $$g$$ of the Galois field $$\mathrm{GF}(q^m)$$ with degree-$$m$$ irreducible polynomial $$f(x)$$ over $$\mathrm{GF}(q)$$. is_primitive_element(element, irreducible_poly) Determines if $$g$$ is a primitive element of the Galois field $$\mathrm{GF}(q^m)$$ with degree-$$m$$ irreducible polynomial $$f(x)$$ over $$\mathrm{GF}(q)$$.

Last update: Jul 12, 2022