Algebraic Number Theory package

## Algebraic Number Theory package

Louis Abraham and Yassir Akram

### Installation

or get the development version with:

### Features

### Jacobi symbol

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

### Solovay-Strassen primality test

 pycon >>> 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

 pycon >>> 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

 pycon >>> 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

 pycon >>> 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

 pycon >>> 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

 pycon >>> 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

 pycon >>> 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)