Tagged Questions

2
votes
0answers
31 views

Find the factorization of the polynomial as a product of irreducible [duplicate]

Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $R[x]$ and $C[x]$ Testing with the simplest possible root in this case, $P(1)=0$ Applying the ...
4
votes
2answers
96 views

Irreducibility of $x^n-x-1$ over $\mathbb Q$

I want to prove that $p(x):=x^n-x-1 \in \mathbb Q[x]$ for $n\ge 2$ is irreducible. My attempt. GCD of coefficients is $1$, $\mathbb Q$ is the field of fractions of $\mathbb Z$, and $\mathbb Z$ ...
2
votes
2answers
64 views

What is “prime factorisation” of polynomials?

I have the following question: Find the prime factorisation in $\mathbb{Z}[x]$ of $x^3 - 1, x^4 - 1, x^6 - 1$ and $x^{12} - 1$. You will need to check the irreduciblity in $\mathbb{Z}[x]$, of ...
2
votes
1answer
42 views

Can the Euclidean algorithm fail by not terminating in non Euclidean domains?

Is it possible for the Euclidean algorithm to fail by not terminating in finite time in non-Euclidean domains? In $\mathbb{Z}[X]$ it can fail by going out of the ring, ie one gets a non integer ...
3
votes
1answer
87 views

Classifying all ideals of a lattice $\mathbb{Z}[\sqrt{-d}]$

In Artin's Algebra he presents a method (that I am sure I am butchering) for classifying ideals of a given lattice $\mathbb{Z}[\sqrt{-d}]$ by taking any ideal $I$, choosing an element of minimum norm ...
1
vote
2answers
233 views

Factoring polynomials of degree 6 in 2 ways.

Let $P(x)$ be an integer polynomial of degree $6$ that is irreducible over the integers. $P(x) = x^6 + (A+a) x^5 + (B+ aA+ b) x^4 + (C+aB+bA +c) x^3 + (aC +bB +cA) x^2 + (bC+cB) x + cC = x^6 + ...
3
votes
2answers
141 views

Factoring in Z3[x]

I need to factor $x^6+x^4+x^2+1$ into irreducible parts in $Z_3[x]$. Obviously this polynomial reduces to $(x^4+1)(x^2+1)$ which is irreducible in $Z[x]$, but I'm not sure how to confirm that it's ...
0
votes
1answer
80 views

About Rings where the elements can be factored like $a^2 b$ … in multiple ways.

Im looking for non-UFD rings such that factoring of any element of that ring into irreducibles leads to either all factorizations squarefree or all factorizations squareful. Thus let $n$ be an ...
0
votes
0answers
95 views

Factoring polynomials $f(g(x))$ over extension fields.

This question is a variation on another one : related question Let $f(x)$ and $g(x)$ be polynomials with integer coefficients and discriminant of $f(x)$ > discriminant of $g(x)$ , which are ...
1
vote
1answer
101 views

Factoring polynomials of degree $a p^b$ over extension fields.

Let $f(x)$ be an irreducible polynomial with integer coefficients, which is irreducible over $\mathbb{Z}$ and has degree $a p^b > p$ with $p,a,b>0$ and $p$ a prime. It appears that $f(x)$ ...
0
votes
2answers
72 views

Find a prime number $p$ so that $f = \overline{3}x^3+ \overline{2}x^2 - \overline{5}x + \overline{1}$ is divided by $x-\overline{2}$ in $\mathbb Z_p$

Let $f = \overline{3}x^3+ \overline{2}x^2 - \overline{5}x + \overline{1}$ be defined in $\mathbb Z_p$. Find a prime number $p$ so that $f$ can be divided by $g = x-\overline{2}$, then factorize $f$ as ...
7
votes
3answers
107 views

How to see that the shift $x \mapsto (x-c)$ is an automorphism of $R[x]$?

In the process of studying irreducibility of polynomials, I encountered the criterion that $p(x)$ is irreducible if and only if $p(x-c)$ is irreducible. When trying to determine what properties of the ...
5
votes
1answer
139 views

Is $x^n+px+p^2$ irreducible in $\mathbb{Z}[x]$?

If $p\in\mathbb{N}$ is a prime, is $x^n+px+p^2$ irreducible in $\mathbb{Z}[x]$? I've proved that any non-unit factor in $\mathbb{Z}[x]$ must have degree at least 2. Eisenstein's criterion doesn't ...
5
votes
2answers
161 views

elegant way to show $P= t^{1024}+t+1$ is reducible in $\mathbf{F}_{2}[t]$

This is homework exercise: $$P=t^{1024} + t + 1 , R = \mathbf{F}_{2}[t] \Rightarrow P \ \text{reducible in R}$$ I wanted to show this analogous to how a book shows it (book shows it with other ...
25
votes
4answers
737 views

Fermat's Last Theorem and Kummer's Objection

In 1847 Lamé had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do ...