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2017-12-04 03:08:48 -0500 asked a question How to visualise complex functions on a disk?

Let $f$ be a function on the unit disk $\mathbb{D}$. I want to look at the images of $f(\mathbb{D})$?

How to do this?

How to see the contours of $|f(z)|$?

How to see the argument (if possible)?

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2016-11-15 02:41:15 -0500 asked a question Trying to find prime factorization of ideals in number fields

Let $L=\mathbb{Q}(\sqrt{-5}, i)$ and $K=\mathbb{Q}(\sqrt{-5})$. Let $O_K$ and $O_L$ be the rings of algebraic integers of $K$ and $L$. It can be checked that

$$2O_K=\langle 2, \sqrt{-5}+1\rangle^2 $$

I want to find the factorization of the ideal $\langle 2, \sqrt{-5}+1\rangle O_L$ in $O_L$ ?

The problem I am having is this. I don't know the syntax for the ideal generated by $\langle 2, \sqrt{-5}+1\rangle O_L$.

What to do ?

2016-11-07 09:50:42 -0500 asked a question Is there any way to find decomposition group and ramification groups

Let $L/K$ be a Galois extension of number fields with Galois group $G$. Let $O_K$ and $O_L$ be the ring of algebraic integers of $K$ and $L$ respectively. Let $P\subseteq O_K$ be a prime. Let $Q\subseteq O_L$ be a prime lying over $P$.

The decomposition group is defined as $$D(Q|P)=\lbrace \sigma\in G\text{ }|\text{ }\sigma(Q)=Q\rbrace$$

The $n$-th ramification group is defined as $$E_n(Q|P)=\lbrace \sigma\in G:\sigma(a)\equiv a\text{ mod } Q^{n+1}\text{ for all } a\in O_L\rbrace$$

I want to compute the decomposition group and ramification groups of the cyclotomic field $\mathbb{Q}(\zeta)$ over $\mathbb{Q}$ where $\zeta$ is a root of unity.

How to do this ? Any idea ?

2016-10-31 01:51:08 -0500 commented answer Finding prime factorization of ideals in number rings

The discriminant of $K=\mathbb{Q}(\sqrt{2}+i)$ is $256$. As $3\nmid 256$, the ideal $\langle 3\rangle$ should remain inert in $O_K$. So how come it splits in $O_K$ ?

2016-10-22 02:32:15 -0500 commented answer Finding prime factorization of ideals in number rings

What is a ?

2016-10-21 08:28:30 -0500 asked a question Finding prime factorization of ideals in number rings

Let $K$ be a number field and $O_K$ its ring of algebraic integers. Let $p\in\mathbb{Z}$ be a rational prime. I want to find the factorization of the ideal $pO_K$. What is the syntax for this ?

For clarity, I request you to demonstrate with an example (say $K=\mathbb{Q}(\sqrt{2}+i)$ and $p=2$ and $p=3$).

2016-10-20 16:21:28 -0500 commented question irreducibility of a polynomial

thanks @slelievre

2016-10-18 06:21:16 -0500 marked best answer if loop not working

I have to use an if loop in my program. I am checking the irreducibility of a collection of polynomials. If f(x) is a reducible polynomial, then I want to find its factor. This is my code

v=f.is_irreducible()
if (v==0) f.factor()

But I am getting a syntax error. Can someone help me ?

2016-10-14 09:08:47 -0500 marked best answer how to find minimal polynomial

How to find the minimal polynomial of an element ? Let $\zeta_n$ be a primitive $n$-th root of unity. I want to find the minimal polynomial of $\zeta_n$ over $\mathbb{Q}(\zeta_n+\zeta_{n}^{-1})$. How do I do that ?

2016-10-08 14:54:40 -0500 asked a question How to find the discriminant of a number field ?

I want to find the discriminant of the number field $\mathbb{Q}(\sqrt2)$ and the field $\mathbb{Q}(\sqrt2,i)$ and $\mathbb{Q}(\sqrt3+i)$ ?

How to do this ?

2016-10-01 11:28:42 -0500 asked a question Listing elements of (Z/nZ)*

I want to find all the elements of (Z/nZ)*. Is there a command for that ? For example if the user gives input 8, the the output will be {1,3,5,7}.

2016-09-23 10:41:06 -0500 asked a question Number of factors of a polynomial

Given a polynomial f, the command f.factor() gives the factorization of f. I want to find out the number of factors of f. Is there any command for that ?

The polynomial ring is assumed to be ZZ[x].

2016-09-02 12:33:45 -0500 asked a question multivariate polynomial ring over complex numbers

I want to factorize bivariate polynomials over C. For single variable case we do this as follow:

R=CC[x]
x=R.gen()
f=x^2+1
f.factor()

How to do this for multivariate case ?

2016-09-02 02:10:29 -0500 asked a question finding the order of a matrix in GL(3,Z)

I am working in GL(3,Z). I want to find all the elements with order less than or equal to 6.

I have the follwoing questions

  1. Given a matrix in GL(3,Z), is there a command to find its order ?
  2. Is there a way to find all the matrices of order 6 ? (On a different context, is it possible to find all the elements of a given order in GL(3,F) where F is a finite field ?)
2016-09-02 01:57:13 -0500 edited question Checking conjugacy of two matrices
  1. I want to check whether or not two matrices are conjugate. I need to check conjugacy for two matrices in $SL(2,\mathbb{Z})$. How to do it ? How to do it for any general ring or field ? (Notice that I just want to verify whether two given matrices are conjugate or not, I don't need the actual matrices; meaning that say $X$ and $Y$ are conjugates and $AXA^{-1}=Y$. I just want to know if or not $X$ and $Y$ are conjugates, I don't need the matrix $A$)

  2. Extending the previous question, say $AXA^{-1}=Y$. Then we can solve the system of linear equations $AX=YA$ to find out $A$. But is there any command which can directly find $A$ ?