Skip to content

louisabraham/algnuth

Folders and files

NameName
Last commit message
Last commit date

Latest commit

 

History

53 Commits
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Repository files navigation

travis

Algebraic Number Theory package

Louis Abraham and Yassir Akram

Installation

pip install --upgrade algnuth

or get the development version with:

pip install --upgrade git+https://github.com/louisabraham/algnuth

Features

Jacobi symbol

>>> from algnuth.jacobi import jacobi
>>> jacobi(3763, 20353)
-1

Solovay-Strassen primality test

>>> from algnuth.jacobi import solovay_strassen
>>> p = 12779877140635552275193974526927174906313992988726945426212616053383820179306398832891367199026816638983953765799977121840616466620283861630627224899026453
>>> q = 12779877140635552275193974526927174906313992988726945426212616053383820179306398832891367199026816638983953765799977121840616466620283861630627224899027521
>>> n = p * q
>>> solovay_strassen(p)
True
>>> solovay_strassen(q)
True
>>> solovay_strassen(n)
False

Quadratic forms

>>> from algnuth.quadratic import *
>>> display_classes(-44)
x^2 + 11⋅y^2
2⋅x^2 + 2⋅xy + 6⋅y^2
3⋅x^2 - 2⋅xy + 4⋅y^2
3⋅x^2 + 2⋅xy + 4⋅y^2
>>> display_primitive_forms(-44)
x^2 + 11⋅y^2
3⋅x^2 - 2⋅xy + 4⋅y^2
3⋅x^2 + 2⋅xy + 4⋅y^2
>>> display_ambiguous_classes(-44)
x^2 + 11⋅y^2
2⋅x^2 + 2⋅xy + 6⋅y^2
>>> display(*reduced(18, -10, 2))
2⋅x^2 + 2⋅xy + 6⋅y^2

Real polynomials

>>> from algnuth.polynom import Polynomial
>>> P = Polynomial([0] * 10 + [-1, 0, 1])
>>> print(P)
X^12-X^10
>>> P(2)
3072
>>> P.disc
0
>>> P.sturm() # Number of distinct real roots
3
>>> P.r1 # Number of real roots with multiplicity
12

Modular arithmetic

>>> P = Polynomial([1, 2, 3])
>>> Pmodp = P % 41
>>> print(Pmodp ** 3)
27⋅X^6+13⋅X^5+22⋅X^4+3⋅X^3+21⋅X^2+6⋅X+1
>>> print((P ** 3) % 41)
27⋅X^6+13⋅X^5+22⋅X^4+3⋅X^3+21⋅X^2+6⋅X+1

Polynomial division

>>> A = Polynomial([1, 2, 3, 4]) % 7
>>> B = Polynomial([0, 1, 2]) % 7
>>> print(A)
4⋅X^3+3⋅X^2+2⋅X+1
>>> print(B)
2⋅X^2+X
>>> print(A % B)
5⋅X+1
>>> print(A // B)
2⋅X+4
>>> print((A // B) * B + A % B)
4⋅X^3+3⋅X^2+2⋅X+1

Berlekamp's factorization algorithm

>>> P = Polynomial([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12])
>>> Pmodp = P % 41
>>> print(Polynomial.ppfactors(Pmodp.factor()))
12⋅(X+31)⋅X⋅(X^2+40⋅X+24)⋅(X^2+36⋅X+13)⋅(X^6+34⋅X^5+26⋅X^4+13⋅X^3+25⋅X^2+26⋅X+35)

Unique Factorization of Ideals

>>> from algnuth.ideals import factorIdeals
>>> factorIdeals(Polynomial([4, 0, 0, 1]))
X^3+4 mod 2 = X^3
(2) = (2, α)^3
X^3+4 mod 3 = (X+1)^3
(3) = (3, α+1)^3
X^3+4 mod 5 = (X+4)⋅(X^2+X+1)
(5) = (5, α+4)⋅(5, α^2+α+1)

About

Algebraic Number Theory package

Resources

License

Stars

Watchers

Forks

Releases

No releases published

Packages

No packages published