Ask Your Question

nebuckandazzer's profile - activity

2019-10-03 18:58:34 -0600 received badge  Famous Question (source)
2019-08-29 11:31:32 -0600 received badge  Notable Question (source)
2019-08-29 11:31:32 -0600 received badge  Popular Question (source)
2019-08-29 08:40:10 -0600 received badge  Popular Question (source)
2019-05-14 22:28:55 -0600 received badge  Nice Question (source)
2019-03-10 18:10:26 -0600 received badge  Popular Question (source)
2019-02-06 22:22:10 -0600 received badge  Notable Question (source)
2019-02-06 22:22:10 -0600 received badge  Popular Question (source)
2018-12-27 03:57:17 -0600 received badge  Popular Question (source)
2018-08-18 02:00:18 -0600 received badge  Famous Question (source)
2018-05-23 16:10:34 -0600 received badge  Notable Question (source)
2018-05-23 16:10:34 -0600 received badge  Popular Question (source)
2017-12-30 10:17:52 -0600 received badge  Notable Question (source)
2017-12-25 06:29:31 -0600 received badge  Notable Question (source)
2017-12-06 02:08:21 -0600 received badge  Nice Question (source)
2017-12-04 03:08:48 -0600 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)?

2017-11-26 15:25:52 -0600 received badge  Famous Question (source)
2017-10-07 17:36:17 -0600 received badge  Popular Question (source)
2017-05-03 06:58:24 -0600 received badge  Notable Question (source)
2017-04-06 13:21:59 -0600 received badge  Popular Question (source)
2017-02-01 08:32:53 -0600 received badge  Popular Question (source)
2016-11-15 02:41:15 -0600 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 -0600 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 -0600 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 -0600 commented answer Finding prime factorization of ideals in number rings

What is a ?

2016-10-21 08:28:30 -0600 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 -0600 commented question irreducibility of a polynomial

thanks @slelievre

2016-10-18 06:21:16 -0600 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 -0600 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 -0600 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 -0600 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 -0600 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 -0600 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 -0600 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 -0600 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 $A$ ?

2016-08-31 15:09:08 -0600 edited question what is the equivalent of cin>> in sage

I am familiar with C++. In C++, the syntax to give input is cin>>. What is the equivalent of cin>> in sage ? I am declaring an integer $n$ (in C++ syntax we write int n) and the user will give the value of $n$ when the program is being run.

How do I do that ?

It would be really helpful if you can give an example. (Say, the purpose of the program is to print an integer given by the user. What will be the syntax ?)