3.1.1 rings.integer_ring - The Integer Ring

Module Author: William A. Stein (was@math.harvard.edu)
Section Author: William A. Stein (was@math.harvard.edu)

The integer_ring module defines the following functions:

crt_basis( X, xgcd=None)
Compute and return a Chinese Remainder Theorem basis for the list $X$ of coprime integers. An alternative xgcd function can be passed as the second argument.

factor( n)
Compute and return the prime factorization of the integer $n$, as a list of pairs $(p_i,e_i)$, where each $p_i$ is prime and $n = \prod p_i^{e_i}$.

The integer ring $\mathbf{Z}$ has the following methods:

characteristic( )
Return 0 as a Pythin int.

is_atomic_repr( )
Return True, since elements of the integers do not have to be printed with parenthesis around them, when they are coefficients, e.g., in a polynomial.

is_field( )
Return False.

name( )
Return the string "Z".

random( b)
Return a random integer between $-b$ and $b$, including both endpoints.

zeta( )
Return $-1$, which is the root of unity of largest order in $\mathbf{Z}$. Note, this is not the Riemann zeta function. it is a member function of the IntegerRing class, and should be thought of in the sense of $\zeta_n$, a primitive $n$th root of unity. This function is available for certain other fields, e.g., number fields.

See About this document... for information on suggesting changes.