# Questions tagged [computational-algebra]

Computational algebra is an area of algebra that seeks efficient algorithms to answer fundamental problems concerning basic algebraic objects (groups, rings, fields, etc.). For questions about generic computer algebra systems, use [tag:computer-algebra-systems].

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### Solving a Solvable Polynomial by Radicals (Effectively)

I'm trying to actually write some code (in sage) to take a polynomial $f$ with solvable galois group and compute its roots as nested radicals. Right now I'm just trying to get cyclic extensions to ...
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### GAP on Jupyter Notebook

I wanted to learn GAP as it is certain that it will be helpful in the future for my research. I wanted to try GAP in Jupyter notebook in GitHub. I have tried following the step i.e. launch binder and ...
41 views

### How do I proof Groebner basis existence in $R[x_1, \dots, x_n]$?

In An Introduction To Groebner Bases from Loustaunau, it says that: We further note that the Noetherian property of the ring R, and hence of the ring $R[x_1, \dots, x_n]$, yields to the following ...
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### Least number of digits of the quotient of a base-B integer division

Let a and b be two integers (b is non-zero) of k and l base B digits, respectively. What is the least number of digits the quotient of the division of a and b can have? I am pretty sure the answer is ...
61 views

### Polynomials which are invariant to the cyclic permutation of variables

I'm trying to solve the following problem from this book. I can find the Gröbner basis of $J$ using Buchberger’s algorithm, and so I don't have any problem with the first part of this problem. But my ...
51 views

### Algorithm for expressing Gröbner basis in terms of ideal generators

Given a polynomial ring and an ideal $$A \supset I = (f_1, ..., f_m)$$ there are plenty of implementations of an algorithm (e.g. Buchberger's) that produces a Gröbner basis $$G = (g_1, ..., g_n)$$ and ...
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### Existence of reduced Groebner basis [closed]

In my class notes there is a Theorem saying: Every non zero ideal of the ring $F[x_1,\dots,x_n]$ has a reduced Groebner Basis. Unfortunately there is no proof. Can someone give the proof or refer to ...
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### Learn Algebra with computational applications [Book Recommendation] [closed]

I am studying Linear and abstract algebra and find it a bit too, again "abstract", could someone recommend me a good book so I can learn it through computational applications? I think it ...
21 views

### Relations between Pseudoprimes

good to everyone. I need your help. Does anyone know the relationships between the pseudoprimes ​​of Catalan, Euler-Jacobi, Frobenius, Lucas, Somer-Lucas and Perrin? and with other pseudoprimes? A ...
171 views

### How to find minimal generating sets efficiently?

Let $G$ be a group. A minimal generating set of $G$ is a subset of $R$ of $G$ that generates $G$ such that no proper subset of $R$ also generates $G$. For arbitrary groups is its not true (unlike ...
63 views

### What is the minimal set of comparisons that determines a monomial order?

A monomial order in $k[x_1, x_2, \ldots, x_n]$ for a field $k$ is a relation $\prec$ on the monomials such that: $\prec$ is a total order; if $m_1 \prec m_2$ then $m_3m_1 \prec m_3m_2$ for any three ...
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### Using a lemma to calculate syzygy.

I want to find the syzygies of the following monomial ideal $I = (x_1^4, x_1^3x_2, x_1^2x_2^2, x_1x_2^3, x_2^4)$ in $S = k[x_1, x_2]$. To do this I will use Lemma 15.1 on pg. 322 in Eisenbud "...
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### Normal form and remainder - Groebner Basis

I'm quite confused about the concept of normal form of a polynomial $f$ relative to the ideal $I$. It should be the remainder of the division of $f$ with a Groebner Basis of $I$. But does that mean ...
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### Is there any function in GAP finding all maximal elementary abelian subgroup of a $p$-group $P$?

I know how to obtain all subgroups of a group in GAP. Is there any function in GAP (or algorithm that I could implemnt myself) for obtaining all maximal elementary abelian subgroups of a $p$-group $P$?...
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EDITED QUESTION: To avoid X-Y problem I am going to write my problem down in detail, so plz bear with me. The elliptic curve over $Q$ given by a Weierstrass equation is - $E := y^2 +a_1 xy +a_3 y = x^... 0answers 42 views ### Are there established algorithms for working with towers of low-degree algebraic extensions? I'm interested in doing 'computational ruler-and-compass' construction simulations along the lines of Euclidea and similar tools. Because the constructions can get rather involved, I'd like to be able ... 1answer 118 views ### Decide if certain polynomial is in an Ideal Let$I$be the ideal$ I = (x^3y-x^2y^2,x^3z+z^2yx,x^2-xz) \subset \mathbb{Q}[x,y,z]$. I have to decide if$x$is part of$I$or$\sqrt I$. My first take was computing the Groebnerbasis$G$of$I$by ... 2answers 130 views ### Q: how to describe these results by a descendants tree in gap I wrote an implement to find the "fullyInvariantGroups" in GAP and the results appeared as below: ... 0answers 38 views ### Factoring matrices over$\mathbb{Z}_k$for$k$composite? Say you have some matrix$C\in\mathbb{Z}_k^{n\times m}$, where$k$is composite, and say the rank of$C$is$r$. Moreover, say that you have some prior knowledge that$C$can be written as$C = AB$, ... 1answer 51 views ### How does GAP calculate 2-closure? GAP software has a method for calculating the two closure of a (permutation) group? how does it do that calculation? 1answer 63 views ### How generate a algebraicaly independent set over rational number field? Algebraic independence. In abstract algebra, a subset${\displaystyle S}$of a field${\displaystyle }$L is algebraically independent over a subfield${\displaystyle }$K if the elements of${\...
I am not clear on how why this proof works. I understand that for any $p$ this algorithm gives us back a $\overline{p}$ such that: $p$ and $\overline{p}$ are congruent mod $I$, and only standard ...