# Tagged Questions

94 views

### factoring $x^n+x+1$

Is there a way of factoring a polynomial of the general form $$x^n+x+1$$ in the ring $\mathbb C[x]$ or $\mathbb R[x]$ or $\mathbb Z [x]$ for any $n \in \mathbb N$? (Or perhaps with certain conditions ...
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### Factor $3x^{3/2}-9x^{1/2}+6x^{-1/2}$

I have $3x^{-1/2} (x^2-3x+2)$ However, I just tried to expand, and the answer is not the same as the original question. With fractional exponents I take out the smallest exponent, then I add the ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $P(P(X))-X$
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### 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 ...
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### $gcd(a,b)$ 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### The polynomial $x^{{n-1}} + x^{n-2} + x^{{n-3}} +\dots + x +1$ is reducible when $n$ is composite

Is $P(x)=(x^{{n-1}} + x^{n-2} + x^{{n-3}} +\dots + x +1)$ reducible if $n>1$ and $n$ not prime? If $n-1$ is odd, $(x+1)|P(x)$, so if $n$ is even with $n>2$ I can write $$P(x)=(x+1) Q(x)$$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
### 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 ...