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Questions tagged [irreducible-polynomials]

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

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For what values of $n$ is $x^3+nx+2$ irreducible? [on hold]

As in the title. I have reduced to irreducibility over the integers by Gauss. We know $f=(x+a)(x^2+bx+c)$ and that $a=\pm1,\pm2$. How should I proceed?
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Is $x^p+a$ always irreducible in $\mathbb{F}_p[X]$?

As in the title. My first thought would be something using Fermat's little theorem, but I'm not sure where to go from there.
4
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1answer
83 views

When is $x^3+x+n$ reducible?

I am trying to work out when $f=x^3+x+n$ is reducible over $\mathbb{Q}$, with $n\in \mathbb{Z}$. So far, I have reduced to irreducibility over $\mathbb{Z}$ by Gauss's lemma, and have that $f=(x+a)(x^2-...
1
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1answer
32 views

Prove $x^4 +x+1$ is irreducible in $\mathbb{Z}/2\mathbb{Z}[x]$ [duplicate]

$ x^4 +x+1$ in $\mathbb{Z}/2\mathbb{Z}[x]$ is an irreducible polynomial. So far we have only treated quadratic and cubic polynomials, which are irreducible if they do not have any zeros. However, ...
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3answers
27 views

Isomorphism between $K[X]/P(X)$ and $K(A)$ for $A$ a root of the irreducible polynomial $P(X)$ over $K$

I'm afraid the answer to this question should be clear to anyone who knows a little bit of algebra, field theory, field extensions, and polynomials. In his answer to a question about Galois theory ...
3
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0answers
60 views

determining if quotient ring of polynomials over a finite field is a field or not

I am stuck with this question: "Determine if $GF(2011^2) [x] /<x^4-6x+12>$ is a field or not." I know that since this polynomial ring is defined over a field, I only have to determine if $x^4-...
0
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1answer
13 views

Proving $L$ is the splitting field of $f(x) = \min_Q(ζ_7)$ [duplicate]

Given the extension $L = Q(ζ_7)$ of $Q$ where $ζ_7 = e^{2πi/7}$, I had a small difficulty in first proving $L$ is the splitting field of $f(x) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$ over $Q$ and that ...
0
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1answer
35 views

Monic polynomial with coefficients in $\mathbb F_2$ whose companion matrix is invertible and has largest possible multiplicative order [on hold]

Let $a_0,...,a_{n-1}\in \mathbb F_2$ be such that the companion matrix of the monic polynomial $a_0+a_1X+\cdots+a_{n-1}X^{n-1}+X^n\in \mathbb F_2[X]$ is invertible and has order $2^n-1$ in $\mathrm{GL}...
2
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1answer
20 views

Counting irreducible polynomial of degree 3 over finite fields with certain restriction

I want help in the following counting problem: We know how many irreducible Monic polynomial of degree 3 are there in $\mathbb{F}_{q}[x]$. Now what I want to find is number of monic irreducible monic ...
2
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3answers
84 views

Factor $x^ 5 - x^4 - x^ 2 - 1$ modulo $16$, and over $\mathbb{Q }$

From ARTIN algebra books chapter $12$ question $4.19$: Factor $x^ 5 - x^4 - x^ 2 - 1$ modulo $16$, and over $\mathbb{Q }$ My works : I have check in $\mathbb{Z}_{16}$ as $x^ 5 - x^4 - x^ 2 - 1$ is ...
0
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1answer
26 views

Irreducible over $\mathbb Q[x]$

If $f(x)= x^3-tx-1$, where $t$ is an integer. For which value of $t$ is $f(x)$ irreducible over $\mathbb Q[x]$? As I think was as; in the field of rational polynomials $\mathbb Q[x]$ (i.e., ...
2
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2answers
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How do I prove that $x^6+x^5+x^4+x^3+x^2+x+1$ is irreducible over $\mathbb{Z}_3[x]$?

I am not really sure that the polynomial $x^6+x^5+x^4+x^3+x^2+x+1$ is really irreducible over $\mathbb{Z}_3[x]$, but all my attempts to factor it failed so far. I know I can't use Eisenstein for this ...
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0answers
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Find the condition on $a, b$ such that the field of $x^3+ax+b\in\Bbb{Q}[x]$ has degree of extension 6.

First of all the polynomial $p(x)=x^3+ax+b$ must be irreducible over $\Bbb{Q}$, because if not then the degree of extension of its splitting field will be $1$ or $2$. Now, suppose $\alpha, \beta, \...
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0answers
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How do I show that $\alpha$ is not an element of $F(\beta)$

I am trying to show that $\alpha$ is not an element of $F(\beta)$: My attempt was by contradiction suppose $\alpha \in F(\beta)$ Then $\alpha = h(\beta) $ where $h$ is a polynomial in $F[x]$ Then $...
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0answers
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Q. Irreducibility criteria

Show that $p(x)=x^2+9x+6$ is irreducible in $\mathbb Q[x]$ according to the following criteria: a) Eisenstein's criterion b) Gauss' theorem c) Irreducibility on $\mathbb Z_p$, $p$ prime. $\...
3
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2answers
27 views

Prove that $\frac{(x+2)^p-2^p}{x}$ is irreducible in $\mathbb{Z[x]}$

Prove that $\frac{(x+2)^p-2^p}{x}$ is irreducible in $\mathbb{Z[x]}$ Can I get some help with this one? I mean my gut is obviously telling me to start with a binomial expansion, but I just don't know ...
0
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1answer
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Prove if $(f,g)=1$ , then $(f^m,g^n)=1$

If $(f,g)=1$ , then prove for any $m,n\in \mathbb{Z}^+$, $(f^m,g^n)=1$ . $(f,g)=\gcd(f,g)$. My try: Split $f$ to product of some irreducible polynomial. Like $f=ap_1^{\alpha_1}\cdots p_t^{\alpha_t} $....
4
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1answer
114 views

Producing a field with $7^3$ elements

Producing a field with $7^3=343$ elements. Okay, so if I can find an irreducible polynomial over $\mathbb Z_7$ of degree $3$ then I'll have done it. Now, since it's of degree $3$ all I have to do ...
2
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2answers
46 views

Factor, or decompose, a polynomial related to the Heptadecagon

As the heptadecagon is constructible with compass and straight-edge, this would seem to indicate that the degree 17 polynomial below $$ T_{17}(x)-1 $$ can be expressed in terms of products and ...
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3answers
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Q. Reducible polynomials in $\mathbb{Z}[x]$

Show that $x^3+ax^2+bx+1$ $\in \mathbb Z[x]$ is reducible on $\mathbb{Z}$ if and only if $a=b$ or $a+b=-2$. If it is reducible, then it has root in $\mathbb Z$. Be $u$ the root, so i can write it ...
0
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1answer
14 views

Product of values of the derivative of a polynomial at its roots

Let $K$ be a field and $f(x) \in K[x]$ be irreducible and separable, with distinct roots $\alpha_1, \ldots, \alpha_n$. In my notes for my algebraic number theory class, I see the equality $$ \prod_{i=...
5
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3answers
66 views

Polynomial in $\Bbb Z [X]$ [closed]

Let $P$ be a polynomial in $\Bbb Z [X]$ of degre $n$ such that $|P(x)|$ is a prime number for $2n+1$ different values of x. Prove that $P$ is irreductible over $\Bbb Z [X]$.
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$n$ for which no polynomial of degree $n$ is irreducible in $\mathbb{Z}_n[X]$ [duplicate]

I had an exercise were I had to prove that for each natural $n$ and each prime $p$, there exists an irreducible polynomial in $\mathbb{Z}_p[X]$. But then I was searching for an example were for a ...
0
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1answer
76 views

$f(x)=x^5-6x+2\in\Bbb Q[x]$ is not solvable by radicals implies it cannot have a root that lies in a radical extension of $\Bbb Q$

Consider $f(x)=x^5-6x+2\in\Bbb Q[x]$ not solvable. Show that $f(x)$ doesn't have a root in a radical extension of $\Bbb Q$. I want to prove it by contradiction: Suppose there is a root of $f$, $u\...
2
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1answer
44 views

Proof of a lemma about irreducible fractions

There is a lemma (maybe called Gauss's lemma) that states that if we have a polynomial $p \in Z[X]$: $p$ is irreducible in $Z[X] \iff p$ is irreducible in $Q[X]$ Can anyone help me with the proof of ...
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0answers
27 views

Check whether the polynomial is irreducible or not

Is polynomial $f(X)=X^{p^n}-X$ is irreducible and separable in $\mathbb{F_p[X]}$? I know that derivative of f(X) is $p^{n}X^{p^n-1}-1$ which is 0-1=-1 so it has no multiple root and hence separable; ...
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0answers
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If $f(X,Y) \in k[X,Y]$ has a zero $(at^n,bt^m)$ where $t$ is transcendental over $k$, then $f$ is of type $X^m Y^n - c$

This is exercise 29 in Chapter V (Algebraic Extensions) from Serge Lang's Algebra. Let $f(X,Y)$ be an irreducible polynomial in two variables over a field $k$. Let $t$ be transcendental over $k$, ...
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Find $(a,b)$ such that $x^2-bx-a$ is irreducible and $\begin{bmatrix}0&a\\1&b\end{bmatrix}$ has order $8$

I believe I might be having a bit of trouble with a question that asks me to find the values of $a$ and $b$ such that the polynomial $x^2-bx-a$ is irreducible and the matrix $\begin{bmatrix} 0 & ...
1
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1answer
26 views

How i can reduce this polynomial in $\mathbb{Z}_2 [x]$?

How i can reduce the polynomial $f(x) = x^9+x^8+x^6+x^5+x^4+x^3+1 \in \mathbb{Z}_2 [x]$. I've tried use the reduction modulo a prime, but that don't solve the problem totally.
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Show that the extension of $E / F$ is of Galois if and only if it has a root of the polynomial $f(x)=x^2+x+1$

$F$ has characteristics 2 and $[E:F]=2$ Assume $ a $ a root of $ f $ such that $ f(x)=(x-a)^2 $, this is impossible since it would imply that $x^2+x+1=x^2-2ax+a^2 = x^2+a^2$, since $1 + 1 = 2 = 0 \ ...
3
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1answer
56 views

Bijection polynomial maps on finite field

It's well known that every function on a finite field is a polynomial funcion. Let $\mathbb{F}$ be a field. It's easy to see a linear map $f:\mathbb{F} \mapsto \mathbb{F}$, $f(x) = ax+b$ is a ...
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1answer
20 views

The reducibility of a polynomial in coefficients of integer

This question is from Putnam and Beyond, an example in the section 'Irreducible Polynomial': Let $P(x)$ be an nth-degree polynomial with integer coefficients with the property that $|P(x)|$ is a ...
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1answer
59 views

Existence of polynomials under a determinant constraint

I am having trouble with the following problem: Do there exist polynomials $a(x), b(x), c(y), d(y) \in \mathbb{R}[x]$ for which the following holds? $$ \forall x, y \qquad 1+xy+x^2y^2 = a(x)d(y) -...
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2answers
56 views

Prove that $2x^4+15x^2+10$ is irreducible in $Q[x]$

The problem suggests using a suitable change of variable of the form $y=ax+b$ and using Eisenstein's criterion to show that it's reducible in $\mathbb{Q}[x]$. So, I have several doubts, first of all, ...
2
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1answer
52 views

Proving that $x^{2^n} + 1$ is irreducible in $Q[x]$

I've been working on this and this is my process: I would like to use Eisenstein's criterion so I considered the substitution $y=x-1$. So $$x^{2^n}+1=(y+1)^{2^n}+1=\sum_{k=0}^{2^n}{2^n \choose k}y^{2^...
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0answers
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Doubly transitive Galois group and specializations

I read here in Lemma 1 that if $\alpha$ is a root of $x^n+ax+b $ and $\frac{x^n-\alpha^n}{x-\alpha}=a$ is irreducible over $k(a,b,\alpha) = k(a,\alpha)$ then the Galois group $G$ is doubly ...
2
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1answer
101 views

Irreducibility Over the Rationals (Still Lacking Extra Condition)

Let $p$ be a prime and $g(x)\in\Bbb Z[x]$ be irreducible modulo $p$. Let $f(x) = g^n(x) + ph(x)$ where $n$ is a positive integer and $h(x)\in\Bbb Z[x]$. Given that $g(x)$ and $h(x)$ are relatively ...
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1answer
36 views

If $F$ is a subfield of an algebraically closed field $K$, then the algebraic closure $\overline{F}$ of $F$ in $K$ is also algebraically closed.

I have solved the problem as follows- As per definition, $\overline{F}$ is the subset of $K$ consisting of all the elements that are algebraic over $F$. We have to show $\overline{F}$ is algebraically ...
0
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1answer
36 views

$P(z,w)=\exp(w)-z=0$ polynomial in $z$ and $w$ and irreducible?

Considering the set $$X=\{(z,w)\in \mathbb{C}^2 \mid P(z,w)=0\},$$ $P:\mathbb{C}²\rightarrow \mathbb{C}$ holomorphic in each variable separately, including any polynomial in $z$ and $w$ and $P$ is non-...
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3answers
91 views

How to factor polynomials in $\mathbb{Z}_{n}[x]$

I realize there already is a question almost identical to this (here), but the answers given are a bit too vague. The problem I have is that I'm looking to factor the polynomial $$x^2+23x+18 \text{ ...
1
vote
0answers
46 views

How to determine the stability of the given polynomial?

Given a stable polynomial $\phi(s)=a_0+a_1{s}+a_2{s^2}+a_3{s^3}+\cdots+a_n{s^n}=\phi^{e}(s)+s\phi^{o}(s)$ where $\phi^{e}(s)=a_0+a_2{s^2}+a_4{s^4}+\cdots,~\phi^{o}(s)=a_1+a_3{s^2}+a_5{s^4}+\cdots$, ...
0
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1answer
36 views

unique factorization in polynomial modulo $p^k$

I read such a judgement,it says: If $p$ is a prime integer,then the non-constant polynomials in $\mathbb Z/(p^k)[x]$ have a unique factorization, for positive integer $k$. So how can I prove it? ...
1
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1answer
46 views

Shift Automorphism of Polynomial rings

In order to prove that cyclotomic polynomial $\Phi_p(x)=x^{p-1}+\dots+x+1$ is irreducible in $\mathbb{Q}[x]$ we need to consider "it's shift", namely $\Phi_p(x+1)$ and in this case we can apply ...
8
votes
1answer
275 views

Does there exist irreducible polynomials of each degree over arbitrary field?

Let $F$ be an arbitrary field such that $[\overline{F}: F]=\infty$. Here $\overline{F}$ denotes the algebraic closure. The question in the title of this post can be rephrased as: Question 1. For ...
4
votes
1answer
81 views

Describe $\mathbb{R}[x]/(x^{2}+ax+b)$ where $a,b \in \mathbb{R}$

So obviously using quadractic formula, we have three cases for the roots which depends on the discriminant. I am not sure if I am right on this but, Case 1: $a^{2}-4b>0$. Then we have two ...
1
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0answers
19 views

Does regular Eisenstein's Criterion apply to a GCD domain?

By "regular" Eisenstein I mean: Let $R$ be a UFD with field of fractions $F$. If $f(x)=a_0+\cdots+a_nx^n\in R[x]$ and there exists a prime $p\in R$ such that $p|a_i$ for $0\le i\le n-1$, $p\not\mid ...
2
votes
2answers
49 views

Factoring Quartics (in quadratic form)

I have some quartic polynomial I wish to factor. Here is an example: $x^4 + x^2 + 1 $ I know the answer to this question $ (x^2 + x + 1)(x^2 -x +1) $ We get these 2 irreducible quadratics. I ...
0
votes
1answer
29 views

Irreducible Polynomials of a Ring

In a worked exam the following three quotients rings: \begin{equation} R_1=\frac{\mathbb{Z}_5[x]}{(x^2)},\space R_2=\frac{\mathbb{Z}_5[x]}{(x^2+1)}, \space R_3=\frac{\mathbb{Z}_5[x]}{(x^2+2)} \end{...
3
votes
3answers
67 views

$p^{n-1}x^n - 1$ over $\mathbb{Q}$ for $p$ prime

Consider $f(x) = p^{n-1}x^n - 1 \in \mathbb{Q}[x]$. I want to show that it's irreducible when $p$ is prime. Neither reduction of the coefficients modulo some prime nor Eisenstein seems to work here. ...
1
vote
1answer
33 views

How to obtain the degree of this field extension?

Let $K/F$ be a field extension, $t,w \in K$ such that $t$ is transcendental over $F$ and $w$ is transcendental over $F(t)$. Prove that for any positive integer numbers $n$ and $m$ the equality $[F(t,w)...