Questions tagged [abstract-algebra]
Abstract algebra is the study of algebraic structures, including groups, rings, fields, vector spaces, and the like.
32 questions
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Factor a polynomial over a finite field or the integers
Without using any built-in factoring/polynomial functions, factor a polynomial completely into irreducibles over the integers or a finite field.
Input
Your program/function will receive some prime (or ...
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Cycling with Rubik's
While idly twisting my Rubik's cube around, my son noticed that it kept going back to the solved state. I'm pretty sure he thought this was some sort of voodoo magic at first, but I explained that if ...
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Find the XOR Primes
In this challenge posed by xnor, we were asked to implement XOR multiplication. In this challenge the goal is to find the first n XOR primes. XOR primes are very ...
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Polynomial Long Division
Implement polynomial long division, an algorithm that divides two polynomials and gets the quotient and remainder:
(12x^3 - 5x^2 + 3x - 1) / (x^2 - 5) = 12x - 5 R 63x - 26
In your programs, you will ...
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No strings attached!
Intro
There are 3 nails in the wall. You've got a piece of string that is fixed to the picture frame with both ends. To hang the picture, you entangled the string with the nails. But before letting ...
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Addition on Elliptic Curves
Disclaimer: This does not do any justice on the rich topic of elliptic curves. It is simplified a lot. As elliptic curves recently got a lot of media attention in the context of encryption, I wanted ...
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Modular multiplicative inverse
Your task is to given two integer numbers, a and b calculate the modular multiplicative inverse of a modulo b, if it exists.
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Counting Abelian groups of a given size
Background
Last time, we counted groups of a given size, which is a non-trivial problem.
This time, we'll only count Abelian groups, i.e., groups with a commutative operation. Formally, a group (G, ∗) ...
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Dihedral group D4 composition with custom labels
The dihedral group \$D_4\$ is the symmetry group of the square, that is the moves that transform a square to itself via rotations and reflections. It consists of 8 elements: rotations by 0, 90, 180, ...
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Finite Field Multiplication
Overview
Given the integer representation of three elements in GF(2^64), give the product of the first two elements over GF(2^64) with the reducing polynomial defined as the polynomial m such that m(...
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(-a) × (-a) = a × a
We all know that \$(-a) \times (-a) = a \times a\$ (hopefully), but can you prove it?
Your task is to prove this fact using the ring axioms. What are the ring axioms? The ring axioms are a list of ...
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Fundamental Solution of the Pell Equation
Given some positive integer \$n\$ that is not a square, find the fundamental solution \$(x,y)\$ of the associated Pell equation
$$x^2 - n\cdot y^2 = 1$$
Details
The fundamental \$(x,y)\$ is a pair of ...
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∀ a b. a + b = b + a
This question is a part of the lean LotM.
A ring is a type of structure that takes the rules of addition and multiplication we are familiar with and abstracts them, so we can reason about them. To do ...
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Counting groups of a given size
Groups
In abstract algebra, a group is a tuple \$(G,\ast)\$, where \$G\$ is a set and \$\ast\$ is a function \$G\times G\rightarrow G\$ such that the following holds:
For all \$x, y, z\$ in \$G\$, \$(...
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Compute the inverse of an integer modulo 100000000003
The task is the following. Given an integer x (such that x modulo 100000000003 is not equal ...
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