Often called prime polynomials. Polynomials that have no polynomial divisors.

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-1
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0answers
43 views

The irreducibility of $a^{4n}+b^{4n}$ [on hold]

How to prove that $a^{4n}+b^{4n} $, for any natural number $ n $, is irreducible over the rationals?
1
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0answers
15 views

Moving a polynomial coefficient so it's irreducible

Given a polynomial over $Z$, $P(X)$, I'd like to show that for any $i$ in $N$, there is a whole number $r$ so that $P(x)+r*x^i$ is irreducible. I'd preferably would love a elementary argument so ...
3
votes
0answers
42 views

There is no irreducible polynomial of largest degree in $\mathbf{F}_q[x]$

I am asked to prove or disprove that given a finite field $\mathbf{F}_q$, the ring $\mathbf{F}_q[x]$ contains irreducible polynomials of arbitrarily large degree. I couldn't think of a reason why this ...
2
votes
2answers
48 views

Irreducibility of polynomials in $\mathbf{Z}_p[x]$ - understanding proofs

I am reading through some irreducibility proofs and there's something I don't quite understand: $x^3+2x+1$ is irreducible in $\mathbf{Z}_3[x]:$ no roots in $\mathbf{Z}_3$ and degree $3$ so ...
1
vote
0answers
56 views

proving that $8x^3-6x-1$ is irreducible over $\mathbb{Q}$

When considering the impossibility of trisecting the 60 degree angle, one comes across the polynomial $f(x)=8x^3-6x-1$, which I want to prove is irreducible over $\mathbb{Q}$. I reduced the ...
0
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0answers
10 views

Proof of $x ≤ \max { (k | c_{m1} |^{1/m1} , k | c_{m2} |^{1/m2} , . . . ,k | c_{m2} |^{1/mk})} .$

Can you please give a proof of this lemma : Let $P (X)$ be an univariate polynomial of degree n $: P (X) = X^{n} + c_{1} X^{n-1} + . . . + c_{n}$ with $ c_{n} \neq 0$. Let $c_{m1} , c_{m2} , . . . , ...
3
votes
1answer
23 views

Can every polynomial be moved to a irreducible one?

Given a polynomial $P(x)$ in $Q[x]$, can I always find a rational number, $c$, so that $P(x)+c$ is irreducible in $Q[x]$?
2
votes
1answer
36 views

Study of irreducibility for rings that are not integral domains.

The standard definition of an irreducible element is that an element of an integral domain $D$ is irreducible if to can not be written as the factor of two non-unit elements of the ring. However, I ...
1
vote
4answers
67 views

Is $x^4 + 4$ irreducible in $\mathbb{Z}_5$?

Well, I'm having doubts, isnt that $\mathbb{Z}_5$ has no zero divisors, and now you cant factor $x^4 + 4$ ?
0
votes
1answer
33 views

Proving that polynomial is irreducible over F and classifying its roots in a splitting field

Let $F =\mathbb F_{2}(u)$ be the field of rational functions over the prime field $\mathbb F_{2}$. Prove that $x^2-u$ is irreducible over $F$ and that it has a double root in a splitting field. ...
2
votes
1answer
65 views

$X^n + X + 1$ reducible in $\mathbb{F}_2$

I was told that sometimes in characteristic 2 that $X^n + X + 1$ is reducible mod 2. What is the smallest $n$ where that is true?
1
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3answers
40 views

Is it true that a polynomial is reducible over a field only if the polynomial has a zero in the field?

I am doing some practice problems for abstract algebra and have come across this idea in a couple places, but it seems fundamentally wrong. For example, in order to prove that $f(x) = x^2 + x + 1$ is ...
1
vote
0answers
20 views

Consequence of Krasner's lemma

I'm really stuck on a problem concerning extensions of discrete valuation fields. The exercise is the following (taken by Serre - Local Fields): Let $K$ be a complete discrete valuation field and let ...
0
votes
1answer
35 views

Showing that the polynomial $f = x^4+x^3+1 \in Z_2[x]$ is primitive?

I have shown it is irreducible. I've tried considering $\alpha = \bar{x} \in \mathbb{Z}_2[x]$ s.t. $\alpha^4+\alpha^3+1 = 0$. From my understanding you use the fact that $\alpha^{16-1} = 1$ and hence ...
0
votes
1answer
43 views

If $f(a)=f(a+1)$, then $F$ has characteristic $0$.

Suppose $f\in F[x]$ is irreducible, $E$ is the splitting field of $f$, and for some $a\in E$ we have $f(a)=f(a+1)=0$. Then $F$ has characteristic $0$. I'm not sure how to use the last assumption: ...
4
votes
1answer
75 views

Show an infinite family of polynomials is irreducible

I would like to show that the polynomials \begin{equation} x^n - 2x^{n-1} - 1 \end{equation} are irreducible over $\mathbf{Z}$, whenever $n > 2$. I've used some computer algebra systems to check ...
0
votes
0answers
30 views

Irreducible polynomial

I'm asking please how to prove that $ P(X,Y)=Y^n - R(X)$ is irreducible in $Q[X,Y]$ for any $R$ monic polynomial with $\gcd(n,\deg R)= 1$.
3
votes
1answer
115 views

Checking If a Polynomial is irreducible

Question Check if the polynomial $$ 2X^{34}Y^{34}+(3X^5Y^3+24X^3Y+9X^2)Y^2+3\in\mathbb{R}[X,Y] $$ is irreducible. Problem Normally when I asked a question here I always had some clue to do something ...
0
votes
1answer
40 views

Irreducible Polynomial over $\mathbb{Z}[X,Y]$

I'm trying to see if the following polynomial is irreducible over $\mathbb{Z}[X,Y]$: $P(X,Y)=X^2Y^3+XY^2+XY+8$ Is there any simple algorithme to prove it ? Thanks
2
votes
0answers
24 views

Linear independence of roots

Given an irreducible polynomial $P(x)\in K[x]$ where $K$ is a field, what are the criteria for the roots of $P$ to be linearly independent over $K$? Edit: fixed in response to comments below
0
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0answers
22 views

Testing a polynomial's reducibility

Let $K/F$ be a finite field extension. If you have a polynomial $f(x)=\prod_{r\in R} (x-r)$ who's splitting field is $K$ and $r \notin F, \forall r\in R$( the set of roots), does $\prod_{r\in X} (x-r) ...
0
votes
0answers
21 views

irreducibility over integral domains

A polynomial $f(x)=g(x).h(x)$ over $D$ ,where $g(x)$ or $h(x)$ must be a unit in $D[x]$ and $D$ is an integral domain, then we say that $f(x)$ is irreducible polynomial over $D$. Since $\mathbb Q$ is ...
1
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0answers
42 views

$R_{a} = R[x]/(x)$ isomorphic to $R_{b} = R[x]/(x-1)$

I am looking at the following two rings: $R_{a} = R[x]/(x)$ and $R_{b} = R[x]/(x-1)$. I was told that these two rings were isomorphic, but I don't see why. Is this due to the minimal polynomials? ...
0
votes
2answers
23 views

Irreducibility of Non-monic Quartic Polynomials in Q[x]

Decide whether the following polynomials are reducible or not in the given ring. $f(x)=5x^4+22x^3+35x^2+47x+15$ in Q$[x]$ $g(x)=2x^4-4x^3-12x^2+14x-2$ in Q$[x]$ I am not sure what method to use to ...
0
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0answers
10 views

irreducibility test for multinomials over a finite field

I am working in an algebraic cryptosystem, and I need, in the process, ensure that a 3-variables polynomial in a finite field is irreducible but I can't find a practical method to do that. Do you know ...
2
votes
2answers
42 views

Transforming a Polynomial to Show Irreducibility Using Eisenstein's Criterion

I have a particular polynomial $$z^5-5z^4+30z^3-150z^2+465z-725$$ A quick check in mathematica shows that this polynomial is irreducible over the rationals, however, it does not pass the third ...
1
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1answer
38 views

Are 0 and -1 the only rational periodic solutions of $z_{n}\equiv z_{n-1}^{2}+c$?

Let $c$ be any complex rational number. Let $z$ be a series of polynomials in $c$ defined by $z_{n}\equiv z_{n-1}^{2}+c$ and $z_{0}\equiv0$ The only rational roots of any $z_{n}$ I have been able ...
2
votes
1answer
38 views

Show that every polynomial of degree $1,2,$ or $4$ in $\mathbb{Z}_2[x]$ has a root in $\mathbb{Z}_2[x]/(x^4+x+1)$.

The problem: Show that every polynomial of degree $1,2,$ or $4$ in $\mathbb{Z}_2[x]$ has a root in $\mathbb{Z}_2[x]/(x^4+x+1)$. My attempt: I know that the polynomials $x$ and $x+1$ have ...
2
votes
1answer
41 views

Let $f(x)=x^5+x^2+1$ with $x_1,x_2,x_3,x_4,x_5$ as zeros and …

Let $f(x)=x^5+x^2+1$ with $x_1,x_2,x_3,x_4,x_5$ as zeros and $g(x)=x^2-2.$ Show that $$g (x_1)g (x_2)g (x_3)g (x_4)g (x_5)-30g(x_1x_2x_3x_4x_5)=7$$. I found this question in a local question paper. ...
0
votes
2answers
26 views

Nilpotent elements of polynomial quotient ring [closed]

Let $F$ be a field and let $f\in F[x]$ be an irreducible polynomial. Are the non-units of $F[x]/(f^n)$ nilpotent elements?
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1answer
32 views

Irreducibility of a Polynomial after a substitution

I am trying to determine whether the polynomial $f(x) = x^6 + 34x^4 + 4x^2 + 89 \in \mathbb{Z}[x]$ is irreducible over $\mathbb{Z}$. Eisenstein's criterion doesn't help and I suppose I could determine ...
3
votes
2answers
66 views

Prove that $f(x)$ is irreducible iff its reciprocal polynomial $f^*(x)$ is irreducible.

This is what I'm trying to prove: Let $f(x)\in\mathbb{Q}[x]$ and $\deg(f(x))>1$. Prove that $f(x)$ is irreducible in $\mathbb{Q}[x]$ iff its reciprocal polynomial $f^*(x)$ is irreducible in ...
0
votes
1answer
37 views

Show that a polynomial is still irreducible in a extension field

I have found this question on the Papantonopoulou's Algebra book: Let $f(x)$ and $g(x)$ be irreducible polynomials over a field $F$ with deg$ f(x) = 15$ and deg$ g(x) = 14$. Let $\alpha$ be a ...
1
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0answers
35 views

Irreducible variety

I have the following problem and seems to stuck with some basic understanding of irreducible and/or non-singular varieties. In $\mathbb{P}^3$ we have an irreducible variety $A$ given by two equations ...
0
votes
1answer
16 views

Let P(x,y,z) be an irreducible homogeneous second degree polynomial. Show that the intersection multiplicity of V(P) with any line l is at most 2.

I came across this question in Algebraic Geometry: A Problem Solving Approach: Let P(x,y,z) be an irreducible homogeneous second degree polynomial. Show that the intersection multiplicity of V(P) with ...
0
votes
1answer
37 views

Irreducible question

Can anyone show me how to prove that $y-x^3$ is irreducible in $\mathbb{A}^2(\mathbb{C})$ For my clarity, the questions asks to decompose variety $V(xy^4-x^7y^2) \subset \mathbb{A}^2(\mathbb{C}) $ ...
1
vote
1answer
55 views

How is $x^3 - x^2 - 1$ irreducible in $\mathbb{F}_3$?

My book says that this polynomial is irreducible over the said field. Clearly, the polynomial does not have root $r = 1$ since $(1)^3 -(1)^2 - 1 = -1 \not\equiv 0 \mod 3$. However, for $r = -1$, we ...
0
votes
1answer
38 views

Irreductibility of $x^p+p-1$ in $\mathbb{Q}[x]$ [closed]

Prove that the polynomial $x^p+p-1$ is irreducible in $\mathbb{Q}[x]$, with $p$ prime. I have been stuck on this problem for a while. Could anyone give me a hint?
1
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2answers
26 views

Unclear proof of irreducibility on $x^m+1$

I came across this problem and frankly, it's unclear looking at the solution. Prove $x^m+1$ irreducible in $\mathbb{Q}(x)$ if and only if $m=2^k$ for $k \in \mathbb{N}$ Well the solution's short ...
0
votes
0answers
61 views

Show that $1+X^4$ is reducible over $\mathbb Z_p$ for every prime $p$. [duplicate]

Show that $1+X^4$ is reducible over $\mathbb Z_p$ for every prime $p$. MY ATTEMPT==>I have used Fermat's theorem for this as $X^{p-1}≡1\bmod p$, then this can also be written in the form ...
0
votes
0answers
40 views

Irreducible over $\mathbb Q[x]$ but reducible over $\mathbb F_p$ for all $p$ [duplicate]

To solve this question I have almost finished the proof but I need a little detail to be rigorous. Let $K$ be the prime fields $\mathbb Q$ or $\mathbb F_p$. Prove that $$f(x)=x^4+1\in ...
0
votes
1answer
54 views

Find all the Zeros and their multiplicities of $f(x)=x^5 +4x^4 +4x^3 -x^2-4x +1$ over $\Bbb Z_5$.

Find all the Zeros and their multiplicities of $f(x)=x^5 +4x^4 +4x^3 -x^2-4x +1$ over $\Bbb Z_5$. Firstly,I've found the zeros of $f(x)$,just by simply substituting the elements of $\Bbb ...
0
votes
1answer
48 views

What are the roots of quintics?

I've been teaching myself a bit of Galois theory and from what I understand, arithmetic operations ranging from addition to taking roots are not enough to express all of the roots of a general ...
0
votes
2answers
51 views

Irreducible implies Separable in a Finite Field

Proposition 37 on page 549 of Abstract Algebra, 3rd Ed. by Dummit and Foote claims that irreducible implies separable over a finite field. Suppose $p(x)$ is irreducible over a finite field of ...
1
vote
1answer
25 views

Irreducibility of special cyclotomic polynomial.

I'm trying to show that $f(x)=1+x^p+x^{2p}+\dots +x^{p(p-1)}$ is irreducible over $\mathbb{Q}[X]$. I'm well aware that cyclotomic polynomials are irreducible, however the (many) proofs of this ...
6
votes
4answers
75 views

$R = \mathbb{Z}[\sqrt{-41}]$, show that 3 is irreducible but not prime in $R$

I'm asked to show that 3 is irreducible but not prime in $R = \mathbb{Z}[\sqrt{-41}]$. And if $R$ is a Euclidean domain. To show that it's not prime I have $(1 + \sqrt{-41})(1 - \sqrt{-41}) = 42 = ...
1
vote
1answer
48 views

$I=(f_1, \ldots, f_n)\subset k[x_1, \ldots, x_n]$ with $f_i\in k[x_i]$ irreducible polynomials

Let $A=k[x_1,\ldots, x_n]$ and $I=(f_1, \ldots, f_n)\subset A$ with $f_i\in k[x_i]$ irreducible polynomials. Is it true that $I$ is a maximal ideal in $A$? $I$ is a maximal ideal $\iff$ $1\in ...
-2
votes
1answer
57 views

Why is $\mathbb{Z}[\sqrt{-5}]$ an integral domain? [duplicate]

I could use some help with this. I know that $\mathbb{Z}[\sqrt{-5}]=\{a+b\sqrt{-5\} }|a,b\in\mathbb{Z}\}$. I then put $$0=(a+b\sqrt{-5})(c+d\sqrt{-5})=ac-5bd+(ad+bc)\sqrt{-5}$$ which leaves me with ...
0
votes
0answers
40 views

Exhibit a reducible polynomial of the form $x^p -x-c$ having no roots in a field of characteristic 0

Is it possible for a polynomial, $x^p -x-c$ where $p$ is prime, to be reducible in a field of characteristic $0$, yet have roots in that field? I know for a fact that the general form is true, ...
0
votes
1answer
75 views

If $x^p−x−c$ is irreducible in $F[x]$ then it has no root in the field.

The complete problem appears in Hungerford's Algebra. Let $c\in F$, where $F$ is a field of characteristic $p$ ($p$ prime). Then $x^p−x−c$ is irreducible in $F[x]$ if and only if $x^p−x−c$ has no ...