2
votes
1answer
43 views

Division by factorized polynomials in Macaulay2

I have this problem dividing by factorized polynomials, for example (x_1^4-x_2^4)//(factor(x_1^2-x_2^2)) does not work because the numerator is of "class R" (R is the ring kk[x_1..x_n]) and the ...
2
votes
1answer
41 views

Help with Polynomial Roots Problem

Let's consider the case of two variables, $p\in\mathbb{R}[x,y]$. Suppose I want to find when there is $c\in\mathbb{R}$ such that $$p(x,x)+p(x,c-x)-p(c-x,x)-p(c-x,c-x)=0 \textbf{ for all } ...
1
vote
0answers
199 views

An Algorithm to Find the Generators of the Radical of a Monomial Ideal

Working over $R=\mathbb{C}[x_1,...,x_n]$, I'm given a ring homomorphism with $i\in{1,...,n}$ and $t\in \mathbb{C}$. $\phi_{i,t}(x_j)=x_j$ for $j\neq i$ to themselves. From this I've proven that an ...
2
votes
2answers
100 views

Mutiple root of a polynomial modulo $p$

In my lecture notes of algebraic number theory they are dealing with the polynomial $$f=X^3+X+1, $$ and they say that If f has multiple factors modulo a prime $p > 3$, then $f$ and $f' = ...
4
votes
2answers
225 views

How to prove that $x^4+x^3+x^2+3x+3 $ is irreducible over ring $\mathbb{Z}$ of integers?

Which criterion (test) one can use in order to prove that $x^4+x^3+x^2+3x+3 $ is irreducible over ring $\mathbb{Z}$ of integers ? Neither of Eisenstein's criterion and Cohn's criterion cannot be ...