I believe this is due to number of precision http://doc.sagemath.org/html/en/reference/rings_numerical/sage/rings/real_mpfr.html It might be a good idea to work with RealField

Equivalently, feed the whole expression to $N$

sage: N(integral(x^x,x,1,2)) #2.0504462345347307

Also, the same syntax $N$ exists in SymPy and the equivalent of $f.n()$ in SymPy is $f.evalf()$

The theorem says a prime $p$ ramifies iff $p|\Delta$. However, this does NOT imply that a prime $p$ which does not ramify must inert, because there is a third case in which the prime splits.

Look at problem for section 3.4, or for detailed treatment (Kummer-Dedekind theorem) see Stevenhagen's lecture notes Number Rings

a is the root of definingPolynomial i.e. $f(a) = 0$

Define your number field $\mathbb{Q}(\alpha), \alpha = \sqrt{2} + i$ .

K.<a> = NumberField(definingPolynomial)

In your particular case

Z.<x> = ZZ[] #Makes x lives in Z[x]
K.<a> = NumberField( minpoly(sqrt(2)+i, x))
I = K.ideal(2)
factor(I)
#(Fractional ideal (1/12*a^3 - 1/4*a^2 - 5/12*a + 5/4))^4
#Even fancier
latex(factor(I))
#(\left(\frac{1}{12} a^{3} - \frac{1}{4} a^{2} - \frac{5}{12} a + \frac{5}{4}\right))^{4}

The problem with Light's algorithm is that one needs to find generators before applying it, otherwise it would be more expensive than usual exhaustive search (from a memory point of view). That being said, I think it worth to implement those tests in sage. I might write a function in python which performs two tests based on arguments given.

As mentioned here, there is a probabilistic algorithm which has running time close to $O(n^2)$. I wonder if there is a deterministic algorithm with running time less than $O(n^3)$.

Similar question has been asked but without a solution

I installed sage through ubuntu PPA's and found an ad-hoc solution to " 'all-toolchain'" problem. It should work in your case since Mint is essentially an Ubuntu.

Step 0: Problem

me@myLaptop:~$ sage -i cryptominisat 
make: *** No rule to make target 'all-toolchain'.  Stop.

Step 1: Installing sufficient dependencies:

According to cryptominitsat's page on github, you need to install some packages

$ sudo apt-get install build-essential cmake
$ sudo apt-get install valgrind libm4ri-dev libmysqlclient-dev libsqlite3-dev

Note: I am not sure if it necessary to apply above steps, but it does not solve the problem

Also, install all packages which have in their name sagemath in Synaptic package manager

Check point

Now, if you try:

me@myLaptop:~$ sage -i cryptominisat

you will get a premission error. Even worse, trying sudo it would give you a root user error!

configure: error: You cannot build Sage as root, switch to an unpriviledged user If you would like to try to build Sage anyway (to help porting)

Step 2: Change /usr/ib/sage and its subfolders' owner to you standard user

$\color{red}{It\ is\ safer\ to\ revert\ ownership\ to\ the\ root}$

Execute

sudo chown -R `whoami`:`whoami` /usr/lib/sagemath/

Finally,

 sage -i cryptominisat

Ta da

Not necessarily, you can install it locally. For this particular case, upload seaborn files into your cloud manually, then open terminal and run python setup.py install --user $whoami in the seaborn's folder. Similarly, you can install Ubuntu's packages but using different commands.

I'll try to come back and read your post thoroughly. Meanwhile, this could be of interest http://doc.sagemath.org/html/en/refer...

Note: I have NOT studied cyclotomic fields.

I believe the problem lies in minpoly() function. It has been reported that minpoly sometimes doesn not return an irreducible polynomial.

Also, if you check where f belongs, you will see that it is not in $\mathbb{Q}(\xi)_5[x]$ !.

sage: K.<b>=CyclotomicField(5);
sage: alpha=1+3*b^2;
sage: f=(1+3*b^2).minpoly()
sage: f.parent()
Univariate Polynomial Ring in x over Rational Field
sage: factor(f)
x^4 - x^3 + 6*x^2 + 14*x + 61
sage: G.<x> = K[] #Define a polynomial ring over K
sage: f = G(f) #Coerce f into G
sage: f.parent()
Univariate Polynomial Ring in x over Cyclotomic Field of order 5 and degree 4
sage: factor(f)
(x - 3*b - 1) * (x - 3*b^2 - 1) * (x - 3*b^3 - 1) * (x + 3*b^3 + 3*b^2 + 3*b + 2)
sage: f.is_irreducible()
False

Update: Could you please elaborate which element would you like to append it to $\mathbb{Q}(\xi)_5$? It seems to me that $\alpha$ is already in $\mathbb{Q}(\xi)_5$ but $\sqrt[5]{\alpha}$ is not. If you use fractional expontiation or the function nth_root in minpoly, it will gives you an error. I am afraid that you have to hardcode the equation $x^5 - \alpha$

sage: alpha in K
True
sage: alpha^(1/5) in K 
False
sage: L.<a> = K.extension(x^5 - alpha)
sage: L
Number Field in a with defining polynomial x^5 - 3*b^2 - 1 over its base field

Second update

If you use fractional exponentiation or the function nth_root in minpoly, it will gives you an error.

It seems an issue need to be solved.

by the way, what f = G(f) means?

$f$ was living in $\mathbb{Q}$,i.e. arithmetic on $f$ would be carry over $\mathbb{Q}$, writing f = G(f) tells sage $f$ is an element of $G$

Example:

sage: a = 7
sage: a^2
49

But,

sage: R = IntegerModRing(13)
sage: a = R(a)
sage: a
7
sage: a^2
10
sage: a + 6
0

For more information see

@mirgee Of course, you need to take of each case. I will try to implement this solution and post it here. I think if you manage to find an old version of sage then it is matter of copying and pasting.

It might be interesting that an older version of sage supports addition and multiplication of continued fraction with each other as well with rational numbers. See William Stein's book Elementary Number Theory: Primes, Congruences, and Secrets. Another quick solution(as you mentioned) is to add sage's magical functions _add_ , _mul_, etc. to the class continued_fraction. It might goes as (for sure it is not an optimal solution): def _add_(self, other): check both types return continued_fraction(self.value + other)

@gilieve I'd recommend Jupyter.

Another way to do that is by using SymPy:

from sympy import Sum, Symbol
x = Symbol('x')
s = Sum( 1/(x**2-1), (x, 1, 00))
s.doit()

Give the result[1]:

0

I remember seeing a way to convert SymPy expression to SAGE format, but I don't know how exactly.

[1] I apologize I typed another function in the command line which is $\frac{1}{x^2 -1}$. After realizing that, I tried to evaluate your function but ipython always crash.