2
votes
1answer
83 views

What do we know about $\displaystyle \frac{f}{\gcd(f,f')}$ if $f\in\mathbb{F}_{p^d}[X]$?

Let $\mathbb{K}=\mathbb{F}_{p^d}$ and $f\in\mathbb{K}[X]$ be a non-constant polynomial with the factorization $$f=\prod_{i=1}^nf_i^{k_i}$$ where $f_i\in\mathbb{K}[X]$ is irreducible and ...
0
votes
1answer
32 views

Finding greatest common divisor between two polynomials.

I have the following past exam question: Calculate $\operatorname{gcd}(x^3 + 2x^2 + 2,2x^2 + 1)$ in $\mathbb{F}_3$ Now I haven't encountered this sort of gcd before(usually I am trying to solve ...
2
votes
1answer
84 views

Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$

I’m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results ...
1
vote
2answers
47 views

Factor irreducible polynomial in Z[x] and R[x]

I've got a couple of problems from an old exam in abstract algebra that I have difficulty in understanding. 1) Write the polynomial $2x^3 - 10$ as a product of irreducible elements in ...
14
votes
1answer
512 views

Is factoring polynomials easier than factoring integers? [duplicate]

I was reading the book Algebra: Chapter 0 , by Paolo Aluffi, and came across the following assertion, in page 290, Exercise 5.9: It is in fact much harder to factor integers than integers ...
1
vote
2answers
54 views

What are irreducible factors?

What are Irreducible factors? I have to solve this question: Find the irreducible factors of $x^4 + 5x^3 + 8x^2 + 9x + 10$ in ${\bf Z}_{11} [x]$. I couldn't find any websites that explained ...
1
vote
0answers
52 views

Proof for maximal ideals in $\mathbb{Z}[x]$

I have been trying to prove the following theorem: Every maximal ideal in $\mathbb{Z}[x]$ has the form $(p, f(x))$ where p is prime integer and f is primitive integer polynomial that is irreducible ...
1
vote
2answers
88 views

Isomorphism between Rings $\mathbb{Z}[\frac{u}{v}]$ and $\mathbb{Z}[\frac{1}{v}]$, u,v relatively prime

Let $u$ and $v$ be relatively prime integers, and let $R'$ be the ring obtained from $\mathbb{Z}$ by adjoining an element $\alpha$ with the relation $v\alpha=u$. Prove that $R'$ is isomorphic to ...
1
vote
1answer
62 views

Artin 2nd Ed. Problem 12.5.3

The problem says "Find the generator for the ideal of $\mathbb{Z}[i]$ generated by $3 + 4i$ and $4 + 7i$." I don't understand the question. It asks us to find the generator of the ideal, but then it ...
2
votes
2answers
63 views

Show that $P(X) -X$ divides $P(P(X))-X$

Let $P$ be a polynomial in $R[X]$. Then show that $P(X) -X$ divides $P(P(X))-X$
1
vote
3answers
57 views

Factoring a polynomial in a field into irreducible

Factor $x^3 + 2x + 3$ into irreducible polynomials in $\mathbb{Z} _5 [x]$ This polynomial has 2 zeros mod 5: x = 2 and x = 4. But these only give me a 2 degree polynomial $x^2 - 4$ and I don't know ...
0
votes
1answer
70 views

$gcd(a,b)$ in a UFD subring is not a greatest common divisor in the ring

Give a counterexample that $R$ is a unique factorization domain but not a principal ideal domain, $S$ is a ring containing $R$, such that $a,b\in R$, $gcd(a,b)$ in $R$ is not a greatest common ...
-3
votes
2answers
74 views

Prove that the subgroup of the quotient group is cycling and infinitely generated

$$M = \left\{\,\dfrac{m}{13^n}\biggm| m\in \mathbb{Z}, n\in\mathbb{N} \,\right\}, \quad G = M/\mathbb{Z}$$ Prove that any subgroup $H < G$, $H\neq G$ is cyclic and infinitely generated and that ...
0
votes
1answer
60 views

Let $R$ be an integral domain and $I$ be a prime ideal of $R$. If $R/I$ is a Euclidean domain, will $R$ be a unique factorization domain?

Let $R$ be an integral domain and $I$ be a prime ideal of $R$. If $R/I$ is a Euclidean domain, will $R$ be a unique factorization domain? I have no idea to prove or disprove this... should I prove ...
0
votes
0answers
70 views

The annihilator of finitely generated modules over PID

Let $R$ be a principal ideal domain. Let $M$ be a finitely generated $R$-module. Suppose there exists prime ideal $p$ and integer $i$ such that $p^i=\operatorname{Ann}(M)$. Then prove: (1) there ...
0
votes
1answer
30 views

If $a/s$ is irreducible in $R_{S}$, then $a$ is irreducible in $R$

Let $R$ be a UFD, and let $S$ be a multiplicative subset of $R$ containing the unity of $R$. Show that if $a/s$ is irreducible in $R_{S}$, then $a$ is irreducible in $R$. I showed that if $a$ is ...
1
vote
2answers
209 views

Rabin's test for polynomial irreducibility over $\mathbb{F}_2$

I know that $f(x) = x^{169}+x^2+1$ is a reducible polynomial over $\mathbb{F}_2$. I want to show this using Rabin's irreduciblity test. First, I then have to check if $f$ is a divisor of ...
2
votes
2answers
179 views

Necessary and sufficient condition for $x^n - y^m$ to be irreducible in $\Bbb C[x,y]$

I'm trying to find a necessary and sufficient condition for $x^n - y^m$ to be irreducible in $\Bbb C[x,y]$. What I have now is rather trivial sufficient conditions: if $m\mid n$ then $m=1$ and if ...
1
vote
1answer
104 views

Multivariate polynomial divisibility and Gauss's lemma

Let $\mathbb{F}$ be a field and $A(x,y)$ and $B(x,y)$ be polynomials in $\mathbb{F}[x,y]$. We would like to prove that $A(x,y)$ divides $B(x,y)$. Will the following approach work? We can interpret ...
5
votes
1answer
125 views

The uniqueness of a special maximal ideal factorization

The following problem is from Michael Artin's Algebra, chapter 12, M.6, unstarred: Let $R$ be a domain, and let $I$ be an ideal that is a product of distinct maximal ideals in two ways, say ...
2
votes
2answers
65 views

Nonunits in a Noetherian Domain have an Irreducible Factor

I think I've proven the following statement without using the fact that it is a domain: Prove every nonunit in a Noetherian domain has an irreducible factor. Proof: Suppose we have a ring which ...
7
votes
3answers
212 views

Is $t^4+7$ reducible over $\mathbb{Z}_{17}$?

Is $f=t^4+7$ reducible over $\mathbb{Z}_{17}$? Attempt: I checked that $f$ has not roots in $\mathbb{Z}_{17}$, so the only possible factorization is with quadratic factors. Assuming ...
1
vote
2answers
66 views

Factorizations of $x^2+x$ in $\mathbb Z_6[x]$

So I was looking through my old algebra book and found a question that I can't seem to answer. Find two Factorizations of $x^2+x$ as the product of nonconstant polynomials that are not associates of ...
0
votes
3answers
73 views

Factorize polynomial in $\mathbb R[x]$ and $\mathbb C[x]$

Factorize the polynomial $x^7-7x^6-x^5+7x^4+x^3-7^2-x+7$ So, I have to factorize this in $\Bbb R[x]$ and $\Bbb C[x]$, but when I'm trying to apply the Ruffini schema, I don't know how to put the ...
1
vote
2answers
72 views

$\operatorname{\mathcal{Jac}}\left( \mathbb{Q}[x] / (x^8-1) \right)$

$\DeclareMathOperator{\Jac}{\mathcal{Jac}}$ Using the fact that $R := \mathbb{Q}[x]/(x^8-1)$ is a Jacobson ring and thus its Jacobson radical is equal to its Nilradical, I already computed that $\Jac ...
2
votes
0answers
46 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 ...
2
votes
2answers
125 views

Is the polynomial $x^3 + 2x^2 + 1$ irreducible in $\mathbb{Z}_{17}[x]$?

Is the polynomial $x^3 + 2x^2 + 1$ irreducible in $\mathbb{Z}_{17}[x]$? It seems this polynomial is reducible. How can I factor this? Thanks!
1
vote
3answers
696 views

Finding irreducible factors of a polynomial over $\mathbb{F}_3$

How to find irreducible factors of a polynomial? For example, how to find the irreducible factors of $x^4+1$ over integers mod 3? I have a start but need some help.
3
votes
2answers
144 views

Factoring the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$

I am trying to factor the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$. I am not exactly sure how to go about doing this, and I feel I am missing some theory in the ...
0
votes
2answers
402 views

How to factor a polynomial modulo p?

Is there a general strategy to factoring a polynomial modulo p? I've looked on Google but I've had a hard time finding anything that specifically outlines an approach that I can understand.
3
votes
1answer
140 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 ...
3
votes
2answers
446 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 ...
4
votes
1answer
164 views

Dr. Math on factoring - mistake?

I am reading this article http://mathforum.org/library/drmath/view/75056.html and would like to ask if this section is correct: If it can, then you would have $f(x,y) = g(x,y) * h(x,y)$, ...
0
votes
0answers
157 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
121 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
1answer
68 views

Distinct-degree factorization in finite fields

In $\mathbb{Z}_3$ $$ x^9 : x^4+x^3+x^2+2x+1 = x^5+2x^4+2x^2+2x$$ with remainder of $x$. In $\mathbb{Z}_7$ $$x^7 : x^4+5x^3+x+5 = x^3+2x^2+4x$$ with remainder of $x$. Is this random? Or is there ...
12
votes
5answers
2k views

Is there a simple explanation why degree 5 polynomials (and up) are unsolvable?

We can solve (get some kind of answer) equations like: $$ ax^2 + bx + c=0$$ $$ax^3 + bx^2 + cx + d=0$$ $$ax^4 + bx^3 + cx^2 + dx + e=0$$ But why is there no formula for an equation like $$ax^5 + ...
1
vote
1answer
165 views

How to factor $x^5 - x + 1$

As I understand it $x^5 - x + 1$ is not solvable by radicals. But it splits over $\mathbb{C}$, so how does it factor into linear factors?
1
vote
2answers
147 views

Factoring $x^4z-2z^2-4x^6+x^2z$

We want to factor $8x^4y^4-2y^8-4x^6+x^2y^4 = -2y^8 + (8x^4+x^2)y^4 -4x^6$. We substitute $x^4$ with $z$: Now we want to compute this $8x^4z-2z^2-4x^6+x^2z = -(x^2-2z)(4x^4-z)$ by hand. Therefore we ...
1
vote
2answers
583 views

Factoring multivariate polynomial

I'm trying to factor $$x^3+x^2y-x^2+2xy+y^2-2x-2y \in \mathbb{Q}[x,y].$$ The hint for the exercise is to use the recursive multivariate polynomial form. So I'm using $\mathbb{Q}[x][y]$: $$ x^3 + ...
3
votes
2answers
208 views

Find $X$, $a$ : ALL prime factors of $(X^a - 1)/(X - 1) < X$

where $X$ is an odd prime, and $a$ is an odd integer. For example, let $X = 37$, $a = 3$, we get $$\frac{37^3-1}{36} = 3 \times 7 \times 67.$$ When factoring numbers such as this, I've noticed that ...
1
vote
1answer
146 views

Unique decomposition of a mapping by an equivalence relation

I have a math question from computer science. The following should be a fundamental fact from mathematics. Can you the mathematicins tell me how you would say it in a more elegant way? Given a ...
2
votes
2answers
216 views

Splitting polynomials

I have a polynomial ${\frac{{{{({z^2})}^p} \pm {p^p}}}{{{z^2} \pm p}}}$ where $p$ is an odd prime number, and I know it splits into two factors $$ \sum_{i = 0}^{p - 1} a_i z^i \text{ and } \sum_{i = ...
5
votes
2answers
335 views

Factoring $X^{16}+X$ over $\mathbb{F}_2$

I just asked wolframalpha to factor $X^{16}+X$ over $\mathbb{F}_2$. The normal factorization is $$ X(X+1)(X^2-X+1)(X^4-X^3+X^2-X+1)(X^8+X^7-X^5-X^4-X^3+X+1) $$ and over $GF(2)$ it is $$ ...