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

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3
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0answers
9 views

For any field $k$ and $n>0$: $X^n-a\text{ reducible }\ \Leftrightarrow\ a=b^p\ \vee\ a=-4b^4$.

I'm stuck on the following exercise: Let $k$ a field and $n$ a positive integer. Prove that $X^n-a\in k[X]$ is reducible if and only if there exist a prime $p$ dividing $n$ and a $b\in k$ such ...
7
votes
3answers
130 views

How can I prove irreducibility of polynomial over a finite field?

I want to prove what $x^{10} +x^3+1$ is irreducible over a field $\mathbb F_{2}$ and $x^5$ + $x^4 +x^3 + x^2 +x -1$ is reducible over $\mathbb F_{3}$. As far as I know Eisenstein criteria won't ...
0
votes
1answer
13 views

module isomorphism inbetween two equivalence classes of polynomials

Let $g \in \mathbb{R}[t]$ be a normed irreducible polynomial of degree 2, meaning that $g(t) = (t - \lambda)(t - \overline{\lambda}$) for a $\lambda = a + b i$, with $a, b \in \mathbb{R}$, $b ≠ 0$. I ...
2
votes
2answers
41 views

Show that the ideal $I=\left\langle x_1^2+1,x_2,…,x_n\right\rangle$ is maximal in $\mathbb{R}[x_1,…,x_n]$.

This is an exercise in "Ideals, varieties, and algorithms" by Cox et al. It first asks to show that $I=\left\langle x^2+1\right\rangle$ is maximal in $\mathbb{R}[x]$. I can show it because it is a ...
1
vote
1answer
52 views

Showing a polynomial is irreducible over $\mathbb{C}[x,y]$

Given $m,n \in \mathbb{N},$ how can I show that the polynomial $x^m+y^n-1$ is irreducible in $\mathbb C[x,y]$? I'm given the following hint, but I don't follow. Note: I know Eisenstein's ...
2
votes
1answer
42 views

Irreducible polynomial modulo 2

I need to prove that polynomial $f(x) = x^{10}+x^{3}+1$ is irreducible modulo $2$. It is irreducible if $f|x^{1024}-x$, isn't it? I can use polynomial long division to check it, but this is not ...
3
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1answer
19 views

Abstract algebra. Proof of: Let $F$ be a finite field and $P$ an irreducible polynomial upon $F$. Then $(F[t]|_{\equiv_P}, + , \ast)$ is a field.

Division of polynomials I put what is unclear to me in between three asterisks bounding the unclear lines... $\equiv_{P}$ is defined as ($\forall Q, S \in F[t]$) $Q\equiv_{P} S \iff ...
-1
votes
3answers
81 views

Factor the polynomial [duplicate]

Factor the polynomial $X^3-X+1$ in $F_{23}$ and $X^3+X+1$ in $F_{31}$. How can I know in which way to factor a polynomial mod $p$? Is there some specified method to do that? Thanks.
1
vote
4answers
69 views

Expansion of $x^n-y^n$

Studying polynomials I couldn't find a way to expand $x^n-y^n$ as a product of other polynomials. Now of course we know that $$x^4-y^4=(x^2+y^2)(x^2-y^2)=(x^2+y^2)(x+y)(x-y)$$ and I came up with this: ...
-1
votes
1answer
40 views

Prove the polynomial is irreducible [duplicate]

I tried this problem for a while, but didn't see the application of Eisenstein's irreducibility criterion here. All the coefficients, including the leading coefficient, are equal to 1. p is a prime ...
1
vote
0answers
28 views

Is this polynomial irreducible over the rationals?

Prove (or disprove): Define $T_n(x)$ as the Chebyshev polynomial of the first kind with degree $n$. If $p$ is an odd prime, then $\sqrt{\frac{T_p(x)-1}{x-1}}$ is an irreducible polynomial over the ...
1
vote
1answer
40 views

Proving that a Galois group $Gal(E/Q)$ is isomorphic to $\mathbb{F}_p^\times$

I have seen many textbooks state this result without proof. $``$ If $E$ is the splitting field for the polynomial $f=x^p-1 \in \mathbb{Q}[X]$ where $p$ is prime, then the Galois group ...
3
votes
1answer
49 views

finding all irreducible polynomials over $\mathbb{F}_3$ up to degree $2$, faster method?

I want to find all irreducible polynomials over $\mathbb{F}_3$ up to degree $2$ and I wonder if there's a better method than the following. The polynomials are of form $aX^2 + bX +c$. So I have to ...
0
votes
2answers
23 views

Let $K$ be a field and let $p(x)\in K[x]$ be an irreducible polynomial of degree $d$. Let $L = K[x]/p(x)$. Prove that $[L:K] = d$.

I'm not sure where to go with this question. I know that $K[x]/p(x)$ is a field since p$(x)$ is irreducible means it is maximal in $K[x]$.
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0answers
24 views

Does $kp +1$ form of divisors of :$ p^n+1 $ , for $n =2k >2$ and $1<kp +1<p^n+1$?

I would be interest to know if $kp +1$ can be the form of divisors of $p^n+1$ , for every even $n >2$ and : $1<kp +1<p^n+1$ How do I prove or disprove it ? Note :$k>0$: is integer ...
1
vote
2answers
24 views

Constructibility of roots of a polynomial

I`m trying to decide if the roots of the polynomial $f(x) = x^4+x^3-2x^2 +x +1$ is constructible. My first thought was to show that the polynomial f is irreducible in $\mathbb{Q}$ then for any root ...
1
vote
3answers
55 views

Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$ [duplicate]

Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$ Attempt. Since $x^2+x+1$ is of degree 2, it is reducible iff it has a root in $\mathbb{F}_p$. It has a root in ...
0
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2answers
48 views

Cases where irreducibility of polynomials over $\mathbb F \iff$ it has no roots in $\mathbb F $

This is question is in light of comments in this question and another question I asked few days back. Quoting Hayden from the first link, You can use Rational Root Theorem to show a polynomial is ...
2
votes
1answer
52 views

Irreducibility of polynomials $x^{2^{n}}+1$

I would like to if the polynomials of the form $x^{2^{n}}+1$ are irreducible over $\mathbb{Q}$ and in that case if there is some "easy" proof for that (where easy means not using a big theory like ...
0
votes
1answer
17 views

multiple roots of irreducible polynomial 2

let say we have an irreducible polynomial over field $F$. I need to prove that all roots of f have the same multiplicity. I know that if $\text{Ch}(F)=0$ so this is easy but I don't know what to do ...
0
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1answer
32 views

Solve $\int (y^4+by^2+c)^{-1/2}dy=x$ as y is a real valued function of x.

Solve $\int (y^4+by^2+c)^{-1/2}dy=x$ as y is a real valued function of x. Here $b$ and $c$ are real constants so that $c>b^2/4$.
5
votes
3answers
120 views

Showing that $x^4 -2x^2 +8 x+1$ is irreducible over $\Bbb Q$

I want to show that the polynomial $$f(x)= x^4 -2x^2 +8 x+1$$ is irreducible over $\Bbb Q$. I've proved it by a long method, but I need an easy and short method. I've try to put $x=t+1$, but this ...
0
votes
1answer
37 views

Irreducibility of a cubic polynomial

Let $f(x)=x^3+2x^2+x-1$. Then over which of the following fields $k$ is $f$ irreducible? $k=\mathbb{Q}$ $k=\mathbb{R}$ $k=\mathbb{F}_2$ $k=\mathbb{F}_3$ My Attempt: (2) $f$ is ...
8
votes
1answer
131 views

Proving that a polynomial of the form $(x-a_1)\cdots(x-a_n) + 1$ is irreducible over $\mathbb{Q}$

I want to prove that for any set of distinct integers $a_1,\ldots,a_n$, the polynomial $$h = (x-a_1)\cdots(x-a_n) + 1$$ is irreducible over the field $\mathbb{Q}$, except for the following special ...
5
votes
1answer
72 views

Show $x^p-t$ has no root in the field $\mathbb{F}_p(t)$

I don't think I fully understand. Let's say there is a root $x_0 \in K=\mathbb{F}_p(t)$, where $p$ is a prime number. Then $x_0 = \frac{P(t)}{Q(t)}$ for some polynomials $P,Q \in \mathbb{F}_p[t]$. ...
6
votes
6answers
135 views

Show that $x^{3}-3$ irreducible over $\mathbb{Q}(\sqrt{-3})$

Is there a slick way to show that $x^{3}-3$ is irreducible over $F= \mathbb{Q}(\sqrt{-3})$? What I did seems kind of convoluted (showing directly that there is no root in F). Thanks
3
votes
2answers
28 views

irreducibility of polynomials made by perturbation from a polynomial

Suppose $f(x)\in\mathbb{Z}[x]$ with $\text{deg}f=2n,n\in\mathbb{Z_+}$ and $f_m(x):=f(x)+ mx^n $ for each integer $m\in\mathbb{Z}$. Let us define a number $P_f$: ...
0
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0answers
28 views

factorization of a cubic polynomial in Z_(p)

here is my exercise (this is training before the exam in a few weeks...): "Consider the polynomial $P(X)=X^3-X^2-1$. What is the factorization of $P$ in $\mathbb{Z}_{p}[X]$?" What I have done so far ...
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2answers
73 views

Proof for quotient polynomial rings equivalent to field extension

I am predominantely looking for a proof, I have seen in my books and around but seem to have a hard time finding that if we let $\alpha_1,\alpha_2,...,\alpha_n$ be the roots of the minimal polynomial ...
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2answers
141 views

How to find the irreducible polynomial?

It is giving me a lot of trouble, and I'm beginning to think it's not possible. Find $\operatorname{irr}(2\sqrt{2} + \sqrt{7})$. I start like this: $x = 2\sqrt{2} + \sqrt{7}$ I have squared ...
4
votes
1answer
84 views

Irreducibility of $X(X-3)(X-\alpha)(X-\beta) + 1$

I'm trying to solve the following exercise: Show that for $\alpha,\beta\geq 3$, the polynomial $f = X(X-3)(X-\alpha)(X-\beta) + 1\in\mathbb Z[X]$ is irreducible. It is straightforward to check ...
1
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1answer
46 views

Construction of addition and multiplication table for GF(4)

I am dealing with finite fields and somehow got stuck. The construction of a prime field $GF(p), p \in \mathbb{P}$ is pretty easy because every operation is modulo p. In other words $GF(p)$ contains ...
0
votes
1answer
16 views

Is there a more efficient way of counting the number of reducible polynomials?

Consider the set of polynomials with degree $n\leq 3$ with coefficients in $\mathbb{Z}_3$. A problem that I am working on is to determine the number of irreducible degree 3 monic polynomials. My ...
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1answer
42 views

When does reducibility over $\mathbb{Z}_n$ imply reducibility over $\mathbb{Z}$

We know that If a polynomial $p$ is reducible over $\mathbb{Z}$, then it is reducible over $\mathbb{Z}_n$, but reducibility over $\mathbb{Z}_n$ doesn't always imply reducibility over ...
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1answer
11 views

Factorizing polynomials: How to calculate $g_{(X)}\in F_{q}[x]$ if we have $f_{(x)} = g_{(x)}^p$

How do I calculate $g_{(X)}\in F_{q}[x]$ if we have $f_{(x)} = g_{(x)}^p$ and $p$ is the characteristic of the field $F$? This problem arises from the factorization of a polynomial into irreducible ...
4
votes
0answers
50 views

Existence of Jordan decomposition over finite field

Prove that over finite field $\mathbb F$ exists additive Jordan-Chevalley decomposition: for all matrix $M$ there are semisimple matrix $M_{s}$ and nilpotent matrix $M_{n}$ such that $M=M_{s}+M_{n}$. ...
2
votes
1answer
36 views

If some non-primitive polynomial in $\mathbb Z[x]$ is irreducible over $\mathbb Q$, does this imply it is irreducible over $\mathbb Z$?

I know from a lemma in Herstein that if a primitive polynomial in $R[x]$ is irreducible in its field of quotients $F[x]$, then it is irreducible in $R[x]$. But, if some non-primitive polynomial in ...
6
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1answer
54 views

Gauss's lemma: More than a stepping stone on the way to proving $R[x]$ is a UFD when $R$ is?

I'm reviewing my abstract algebra a bit. Currently looking at UFDs. In this context, Gauss's lemma (or part of it, at least) says that the product of two primitive polynomials over a UFD is primitive. ...
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1answer
139 views

Irreducible Polynomials over Finite Fields [closed]

How would I show that $p(x)=x^5+x^2+1$ is an irreducible polynomial over $\Bbb Z_2=\{0,1\}$.
5
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2answers
114 views

Proving Irreducibility of $x^4-16x^3+20x^2+12$ in $\mathbb Q[x]$

Trying to prove that the following polynomial is irreducible in $\mathbb Q[x]$: $x^4-16x^3+20x^2+12$ What I have tried: 1.) Eisenstein's Criterion, but there exists no suitable prime. 2.) ...
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vote
4answers
151 views

Show a polynomial is irreducible mod 29

Is there an easy way to see that the polynomial $x^2 + 3x + 10$ is irreducible modulo 29 without having to go through each element 0,1,..,28 and check for roots?
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1answer
32 views

Exercise: splitting field, showing that it splits

I need help with this exercise: Let $\alpha$ be a zero of $x^3+x^2+1$ in $\mathbb{Z}_2$. Show that $x^3+x^2+1$ splits in $\mathbb{Z}_2(\alpha)$. [Hint: There are eight elements in ...
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1answer
27 views

a question about field theory and polynomials

Hello all I was given this question in my field theory class on which I would certainly appreciate the help: I am given a field F of characteristic p ($ ch(F) > 0 $) and this polynomial $ f(x) = ...
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1answer
14 views

Question about irreducible polynomials?

Is this polynomial: $irr(\sqrt{3 -\sqrt{6}}, \mathbb{Q})$ irreducible? Here is what I did $ a = \sqrt{3 -\sqrt{6}}$ $a^2 = 3 - \sqrt{6}$ $a^2 - 3 = -\sqrt{6}$ $(a^2 - 3)^2 = 6$ Our polynomial ...
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votes
0answers
16 views

Showing irreducibility of polynomials of degree 3 over the rationals

Let $\ g = X^3\ -9X + 16 $. Prove that $g$ is irreducible over the rational numbers. So far I have used reduction modulo $5$ and this gives $g_5 = X^3 +X + 1$. Then I get $$ g_5(0) \equiv 1 \pmod5,\\ ...
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2answers
23 views

Finding the conjugates, why can they argue this way?(exercise)

In one exercise I am supposed to find the conjugate of $\sqrt{2}+i$ over $\mathbb{Q}$. I found the answer by finding irr$( \sqrt{2}+i,\mathbb{Q})$, and then solving the polynomial finding all the ...
1
vote
1answer
57 views

Why can't Eisenstein Criterion be used for certain polynomials (to show that it's irreducible over $\mathbb{Q}$)?

Why can't Eisenstein's Criterion be used to show that $$4x^{10} - 9x^{3} + 21x - 18$$ is irreducible over $\mathbb{Q}$? I mean even if we were to apply Eisenstein here, there doesn't exist a prime ...
0
votes
1answer
22 views

Why generator polynomial of $GF(2^m)$ are irreducible?

Why generator polynomial of the cyclic group $GF(2^m)$ are irreducible?
5
votes
0answers
30 views

Why Fibonacci LFSR random number generation works?

If I use primitive polynomial of GF($2^m$) in Fibonacci LFSR, it is generating all m-length binary combinations. But, I cannot understand why this should happen. I am not getting any mathematical ...
1
vote
1answer
26 views

Irreducible quadratic “within” reducible quadratic

If we have a reducible quadratic function \begin{equation*} P(x)=a_1x^2+b_1x+c_1=(rx-x_1)(tx-x_2),~x_1,x_2,r,t\in\mathbb{Z}, \end{equation*} does there exist another irreducible quadratic function ...