Algebraic Number Theory 10/10/2012
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Basics
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{{{id=129|
///
}}}
{{{id=125|
2+2
///
4
}}}
{{{id=5|
2*2
///
4
}}}
If you are familiar with Python, Sage does a bit of preparsing to make the syntaxed more "math friendly".
So 2^4 does mean $2^4$.
{{{id=6|
2^3
///
8
}}}
{{{id=126|
2*(4-5)^5-3
///
-5
}}}
x is predefined to be a variable
{{{id=7|
x^2
///
x^2
}}}
{{{id=127|
y
///
y
}}}
{{{id=9|
y = var('y')
///
}}}
{{{id=10|
x+y
///
x + y
}}}
Even though x is a predefined variable, you can assign objects to it.
But then it's no longer a variable, so be careful.
{{{id=11|
x = 2
///
}}}
{{{id=12|
x+y
///
y + 2
}}}
{{{id=13|
x = var('x')
///
}}}
{{{id=14|
x+y
///
x + y
}}}
{{{id=15|
f = x^3+x-2
///
}}}
{{{id=130|
f
///
x^3 + x - 2
}}}
{{{id=16|
f(x=2)
///
8
}}}
{{{id=131|
x
///
x
}}}
{{{id=17|
g = x*y+y
///
}}}
{{{id=18|
g(y=3)
///
3*x + 3
}}}
{{{id=132|
plot(f,(-10,10),color = 'purple')
///
}}}
{{{id=115|
plot(f,(-5,5),color = 'green')+plot(g(y=3),(-5,5),color = 'red')
///
}}}
Lists, loops, and Indexing
Python indeces start at 0! (Not 1!)
{{{id=109|
l = [i for i in range(10)]
l
///
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
}}}
{{{id=116|
len(l)
///
10
}}}
{{{id=111|
for i in range(10,15):
print i
///
10
11
12
13
14
}}}
{{{id=134|
fac = [(next_prime(10^i)+1).factor() for i in range(10,15)]
///
}}}
{{{id=112|
for i in l:
if i%2 == 0:
print i
else:
print 'not even'
///
0
not even
2
not even
4
not even
6
not even
8
not even
}}}
{{{id=117|
m = [i for i in range(10,14)]
print m
///
[10, 11, 12, 13]
}}}
{{{id=118|
m = [i for i in range(1,15)]
///
}}}
{{{id=119|
m[0], m[-1]
///
(1, 14)
}}}
{{{id=120|
m[1],m[-2]
///
(2, 13)
}}}
Matrices
{{{id=19|
A = matrix([[-1,2,3],[2,1,3],[3,0,0]]); A
///
[-1 2 3]
[ 2 1 3]
[ 3 0 0]
}}}
{{{id=136|
latex(A)
///
\left(\begin{array}{rrr}
-1 & 2 & 3 \\
2 & 1 & 3 \\
3 & 0 & 0
\end{array}\right)
}}}
{{{id=21|
A.echelon_form()
///
[1 0 6]
[0 1 0]
[0 0 9]
}}}
{{{id=22|
A.base_ring()
///
Integer Ring
}}}
{{{id=23|
B = matrix(QQ,[[-1,2,3],[2,1,3],[3,0,0]])
///
}}}
{{{id=24|
B.echelon_form()
///
[1 0 0]
[0 1 0]
[0 0 1]
}}}
{{{id=137|
A.change_ring(QQ).echelon_form()
///
[1 0 0]
[0 1 0]
[0 0 1]
}}}
{{{id=25|
B.base_ring()
///
Rational Field
}}}
{{{id=26|
S, U, V = A.smith_form()
///
}}}
{{{id=27|
S, U, V
///
(
[1 0 0] [ 0 1 0] [ 0 1 -3]
[0 1 0] [ 1 -2 0] [ 1 4 -9]
[0 0 9], [ 3 -6 -1], [ 0 -2 5]
)
}}}
{{{id=138|
U*A*V == S
///
True
}}}
{{{id=139|
A.base_ring()
///
Integer Ring
}}}
{{{id=28|
A,B
///
(
[-1 2 3] [-1 2 3]
[ 2 1 3] [ 2 1 3]
[ 3 0 0], [ 3 0 0]
)
}}}
{{{id=140|
B.base_ring()
///
Rational Field
}}}
{{{id=29|
C = A.change_ring(RR); C
///
[-1.00000000000000 2.00000000000000 3.00000000000000]
[ 2.00000000000000 1.00000000000000 3.00000000000000]
[ 3.00000000000000 0.000000000000000 0.000000000000000]
}}}
{{{id=141|
C.change_ring(ZZ)
///
[-1 2 3]
[ 2 1 3]
[ 3 0 0]
}}}
{{{id=121|
M = identity_matrix(5); M
///
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
}}}
{{{id=122|
N = zero_matrix(5,5); N
///
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
[0 0 0 0 0]
}}}
Useful things:
tab completion
?
??
search_src('stuff to look for')
{{{id=142|
///
}}}
{{{id=123|
search_src('number field')
///
}}}
Fields, Rings and Ideals
Number Fields
Basic examples, $\mathbb{Z}$ and $\mathbb{Q}$
{{{id=143|
QQ.zero_element()
///
0
}}}
{{{id=144|
QQbar.ideal(7)
///
Principal ideal (1) of Algebraic Field
}}}
{{{id=30|
ZZ.is_noetherian()
///
True
}}}
{{{id=99|
///
}}}
{{{id=40|
I = ZZ.ideal(18)
J = ZZ.ideal(5)
I,J,I+J
///
(Principal ideal (18) of Integer Ring, Principal ideal (5) of Integer Ring, Principal ideal (1) of Integer Ring)
}}}
{{{id=37|
I.gens(),J.gens(), (I+J).gens()
///
((18,), (5,), (1,))
}}}
Everyone's favorite number field:
{{{id=39|
x = var('x')
F. = NumberField(x^2-x-1)
print F
OF = F.ring_of_integers()
OF.gens()
///
Number Field in a with defining polynomial x^2 - x - 1
[1, a]
}}}
{{{id=149|
OF.basis()
///
[1, a]
}}}
{{{id=150|
F.gens()
///
(a,)
}}}
{{{id=45|
a in OF
///
True
}}}
{{{id=46|
a+2 in OF
///
True
}}}
{{{id=47|
a/2 in OF
///
False
}}}
{{{id=151|
a/2 in F
///
True
}}}
{{{id=48|
(a+2).minpoly()
///
x^2 - 5*x + 5
}}}
Can create and work with ideals as before.
{{{id=54|
I = F.ideal(5)
print I
print I.factor()
///
Fractional ideal (5)
(Fractional ideal (-2*a + 1))^2
}}}
{{{id=55|
J = F.ideal(5).factor()[0][0]; J
///
Fractional ideal (-2*a + 1)
}}}
{{{id=152|
F.ideal(5).factor()[0]
///
(Fractional ideal (-2*a + 1), 2)
}}}
{{{id=56|
F.ideal((7-21*a)*5*(3*a+13)).factor()
///
(Fractional ideal (-3*a + 1)) * (Fractional ideal (3*a + 13)) * (Fractional ideal (-2*a + 1))^2 * (Fractional ideal (7))
}}}
{{{id=49|
K. = QuadraticField(-5)
///
}}}
{{{id=153|
b.minpoly()
///
x^2 + 5
}}}
{{{id=50|
F.class_number()
///
1
}}}
{{{id=51|
K.class_number()
///
2
}}}
{{{id=52|
K.ring_of_integers().gens()
///
[1, b]
}}}
{{{id=53|
b.is_integral()
///
True
}}}
{{{id=58|
L. = NumberField([x^2-2,x^2-x-1])
///
}}}
{{{id=61|
///
}}}
{{{id=103|
OL = L.ring_of_integers()
///
}}}
{{{id=105|
O = L.order([1,a,b,a*b])
///
}}}
{{{id=104|
O == OL
///
True
}}}
{{{id=154|
OL.basis()
///
[(b + 2)*a - 3*b, a - b, -2*b*a + b + 3, (3*b + 5)*a - 8*b - 1]
}}}
{{{id=106|
alpha = a+3*b
///
}}}
{{{id=107|
mList = [alpha*e for e in O.basis()]
mList
///
[(6*b + 3)*a - 7*b - 5, 2*b*a - 3*b - 1, (-5*b - 3)*a + 8*b + 3, (16*b + 8)*a - 21*b - 14]
}}}
{{{id=62|
F.gen()
///
a
}}}
{{{id=155|
d = F.degree()
bas = [F.gen()^i for i in range(d)]
bas
///
[1, a]
}}}
{{{id=156|
M. = L.absolute_field()
///
}}}
{{{id=157|
d = M.degree()
bas = [M.gen()^i for i in range(d)]
bas
///
[1, c, c^2, c^3]
}}}
{{{id=160|
elt = 1+c+3*c^3
///
}}}
{{{id=161|
elt.vector()
///
(1, 1, 0, 3)
}}}
{{{id=162|
///
}}}
{{{id=158|
mbas = M.ring_of_integers().basis()
///
}}}
{{{id=163|
matrix([(elt*m).vector() for m in mbas])
///
[ 32/3 182/3 41 -70/3]
[ 3 19 16 -6]
[ -6 -33 -11 28]
[ 28 162 107 -67]
}}}
{{{id=159|
print L.gens()
print M.gens()
///
(a, b)
(c,)
}}}
{{{id=63|
e1,e2 = F.embeddings(CC)
///
}}}
{{{id=66|
F.gens()[0],e1(F.gens()[0]),e2(F.gens()[0])
///
(a, -0.618033988749895, 1.61803398874989)
}}}
{{{id=64|
c = a+b
///
}}}
{{{id=65|
c.absolute_minpoly()
///
x^4 - 2*x^3 - 5*x^2 + 6*x - 1
}}}
{{{id=67|
c.minpoly()
///
x^2 - 2*b*x + b - 1
}}}
{{{id=68|
OL = L.ring_of_integers()
OL.gens()
///
[(b + 2)*a - 3*b, a - b, -2*b*a + b + 3, (3*b + 5)*a - 8*b - 1]
}}}
{{{id=69|
kL = OL.residue_field(L.ideal(7).factor()[0][0])
///
}}}
{{{id=70|
kL.characteristic()
///
7
}}}
{{{id=71|
L.ideal(7).absolute_norm().factor()
///
7^4
}}}
{{{id=72|
O = L.order([1,a,b,a*b])
///
}}}
{{{id=73|
O.absolute_discriminant().factor()
///
2^6 * 5^2
}}}
{{{id=74|
OL.absolute_discriminant().factor()
///
2^6 * 5^2
}}}
{{{id=76|
O == OL
///
True
}}}
{{{id=77|
OL.basis()
///
[(b + 2)*a - 3*b, a - b, -2*b*a + b + 3, (3*b + 5)*a - 8*b - 1]
}}}
{{{id=78|
On = L.order([2,2*a,2*b,2*a*b])
///
}}}
{{{id=79|
On.absolute_discriminant().factor()
///
2^12 * 5^2
}}}
{{{id=80|
On.index_in(OL)
///
8
}}}
{{{id=81|
CyclotomicField(5)
///
Cyclotomic Field of order 5 and degree 4
}}}
{{{id=82|
F.galois_group()
///
Galois group of Number Field in a with defining polynomial x^2 - x - 1
}}}
{{{id=83|
LG = L.galois_closure('c')
///
}}}
{{{id=84|
LG.absolute_discriminant()
///
1600
}}}
{{{id=85|
c.minpoly()
///
x^2 - 2*b*x + b - 1
}}}
{{{id=87|
(a+b).minpoly()
///
x^2 - 2*b*x + b - 1
}}}
Now a Function Fields Example
First, how one might nievely try. It works, painfully.
{{{id=90|
R. = QQ[]
print R
I = R.ideal([a^2-t, b^2-(t+1)])
print I.gens()
///
Multivariate Polynomial Ring in a, b, t over Rational Field
[a^2 - t, b^2 - t - 1]
}}}
{{{id=91|
Q. = R.quotient(I);
print Q
S. = Q[]
f = (x^2+1)^2 - 2*(t+1)*x^2
f
///
Quotient of Multivariate Polynomial Ring in a, b, t over Rational Field by the ideal (a^2 - t, b^2 - t - 1)
x^4 - 2*t*x^2 + 1
}}}
{{{id=92|
alpha = a+b; alpha
basis = [1,a,b,a*b]; basis
im = [b*alpha for b in basis]; im
///
[a + b, a*b + t, a*b + t + 1, a*t + b*t + a]
}}}
{{{id=93|
# I figured out this matrix by *looking* manually at im above. -- ugly...!
mat = matrix(4,4,[[0,1,1,0], [t,0,0,1], [t+1,0,0,1],[0,t+1,t,0]]);
print mat
f = mat.charpoly(); f
///
[ 0 1 1 0]
[ t 0 0 1]
[t + 1 0 0 1]
[ 0 t + 1 t 0]
x^4 + (-4*t - 2)*x^2 + 1
}}}
{{{id=96|
print alpha^4
print f(alpha)
print (x^2+1)^2 - 2*(t+1)*x^2
///
8*a*b*t + 4*a*b + 8*t^2 + 8*t + 1
0
x^4 - 2*t*x^2 + 1
}}}
Here's how to do this using function fields
{{{id=88|
F. = FunctionField(QQ)
K. = F[]
R. = F.extension(x^2-t)
S. = R[]
T. = R.extension(x^2-(t+1))
print T
print (a+b).charpoly()*(-a+b).charpoly()
# answer appears here.
///
Function field in b defined by x^2 - t - 1
x^4 + (-4*t - 2)*x^2 + 1
}}}
{{{id=98|
///
}}}
{{{id=133|
search_src('contour plot')
///
Search Source Code: "contour plot"
- plot/contour_plot.py
- plot/graphics.py
- server/notebook/interact.py
}}}
{{{id=145|
(19).mod(13)
///
6
}}}
{{{id=146|
k = ZZ.residue_field(131)
///
}}}
{{{id=147|
k.characteristic()
///
131
}}}
{{{id=148|
///
}}}