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

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1answer
13 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 ...
0
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1answer
36 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 ...
0
votes
1answer
10 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
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0answers
36 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
33 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
votes
1answer
45 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. ...
-3
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0answers
37 views

Let $F =\mathbb Z_3$. Give an example of irreducible monic polynomial in F[x]. [closed]

Let $F =\mathbb Z_3$. Give an example of irreducible monic polynomial in F[x]. Then give an extension field $\mathbb Z_3 < H$ and classify according to the fundamental theorem of abelian groups. ...
2
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1answer
103 views

Irreducible Polynomials over Finite Fields [closed]

How would I show that $p(x)=x^5+x^2+1$ is an irreducible polynomial over $\Bbb Z_2=\{0,1\}$.
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2answers
86 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.) ...
1
vote
4answers
147 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
26 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 ...
1
vote
1answer
26 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) = ...
0
votes
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
15 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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vote
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
54 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
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1answer
19 views

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

Why generator polynomial of the cyclic group $GF(2^m)$ are irreducible?
5
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0answers
25 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
25 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 ...
-1
votes
1answer
30 views

Example of a non-primitive but irreducible polynomial

A polynomial $f(x)=a_0+a_1x+......+a_nx^n\in R[x]$ where $R[x]$ is a polynomial ring over a ring $R$ is said to be primitive if $\gcd(a_0,a_1,a_2,......,a_n)$ is a unit. I could find examples of ...
3
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0answers
81 views

proof that $\frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1}$ is irreducible

I am reading the group theory text of Eugene Dickson. Theorem 33 shows this polynomial is irreducible $$ \frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1} \in \mathbb{Z}[x]$$ He shows this polynomial ...
1
vote
1answer
32 views

Minimal polynomials

Can someone explain to me how the minimal polynomials in page 4 of this document are obtained? Please help me. http://web.ntpu.edu.tw/~yshan/BCH_code.pdf It should be something standard about ...
2
votes
1answer
88 views

Reducible polynomials in $\mathbb{Z}[X]$

Let $(a_n)_{n\geq 1}$ be a strictly increasing sequence of integers and $k$ a non-zero integer such that for some $N\in \Bbb Z^+$, the polynomial $$ p_{N}(x)=(X-a_1)(X-a_2)\cdots(X-a_N)+k $$ ...
2
votes
2answers
84 views

Is $x^4+nx+1$ irreducible?

Consider the polynomial $\xi= x^4+nx+1\in \mathbb Z[x]$. Show that if $n=\pm2$ then $\xi$ is reducible and that $n\neq\pm2$ implies $\xi$ is irreducible. I got the answer by writing the ...
0
votes
1answer
20 views

Irreducible polynomial and its roots

Suppose I have a polynomial with rational coefficients which is irreducible over the rational numbers. Let $a$ be one of its roots. Can I express the other roots of this polynomial in terms of $a$ ...
0
votes
3answers
72 views

A transformation of F[x]

I am confused on how to get this problem started. I want to say that they are associated, but I am not sure if that is correct. Let $G$ be the subset of $F[x]$, $F$ a field, consisting of all the ...
2
votes
2answers
41 views

Writing $P_n=\sum_{\sigma \in \mathfrak{S}_n} X^{c_n(\sigma)}$ as irreducible factors in $\mathbb{Q}[X]$.

Let $\sigma \in \mathfrak{S}_n$, denote $\alpha_n(\sigma)$ the number of cycles in the decomposition of product of disjoint cycles. Let $$P_n=\sum_{\sigma \in \mathfrak{S}_n} ...
0
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3answers
40 views

If $p$ is a prime, prove there are exactly $\frac{p^3-p}{3}$ monic irreducible cubic polynomials in $\mathbb{Z}_p[x]$

If $p$ is a prime, prove there are exactly $\frac{p^3-p}{3}$ monic irreducible cubic polynomials in $\mathbb{Z}_p[x]$ I am looking some notes here but don't know in general how to approach this ...
0
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1answer
27 views

How many monic primitive quadratic polynomials are there in $Z_{7}[x]$?

A theorem states that "for each prime p and for each integer $n \ge 1$, there exists a monic irreducible polynomial of degree n in $Z_{p}[x]$". I am not sure if this theorem will help answer my ...
3
votes
2answers
55 views

Prove or disprove $R= \mathbb Q[x]/\langle x^3-x^2+x-1 \rangle$ is an integral domain.

I've got $R$ is not a field since the polynomial is reducible in $ \mathbb Q[x]$. Is it possible to say anything from this?
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2answers
51 views

a question about abstract algebra,the order of $\Bbb Z_{5}[x]/ (x^3+x+1)$

Firstly, I have proven that $x^3+x+1$ is irreducible in $\Bbb Z_{5}[x]$,then how can I know the order of $$\Bbb Z_{5}[x]/ (x^3+x+1),$$ where $(x^3+x+1)$ means the ideal generated by $x^3+x+1$. Can ...
2
votes
0answers
35 views

Irreducible Polynomial-Am I doing this wrong?

Ok,this problem might appear a bit trivial but I have some doubts..If it's not a burden take a look and comment! Let $F$ be a finite field of characteristic equal to $p$ and $ƒ(x)=x^p-α$ $∊F[x]$.Show ...
4
votes
1answer
61 views

For a polynomial $f\in K[x]$, when is there a constant $c\in K$ such that $f+c$ is irreducible?

I was working on a different problem when the following question occurred to me: For a polynomial $f\in K[x]$, is there always a constant $c\in K$ such that $f+c$ is irreducible? Obviously this is ...
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0answers
79 views

How to make addition of two polynomials have no integer root.

Consider I have two degree $d$ polynomials $f_1$ and $f_2$, and they do not have any root in common. I need to compute $f_3=f_1+f_2$, but $f_3$ may have some roots in $R$. So I pick two random ...
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votes
1answer
36 views

What does theorem 23.11 mean from Fraleigh?

I'm reading the theorem, but I don't quite understand it. It says: If f(x) is an element of $Z[x]$ then f(x) factors into a product of two polynomials of lower degrees $r$ and $s$ in $Q[x]$ if ...
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vote
1answer
39 views

Find the roots of the polynomial in $Z_5$

The polynomial is: $2x^{219} + 3x^{74} + 2x^{57} + 3x^{44}$. Find the zeros. Now my first step, which I believe shall be correct is to reduce the exponents of the polynomial in mod 5. Thus: 219 ...
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0answers
43 views

Observation about Polynomials Addition

For two polynomials $f_1=(x-a)(x-b), f_2=(x-a)(x-e)$, if we add them together: $f_3=f_1+f_2$, $f_3$ only has an integer root that is $a$. I observed that it'd possible to make $f_3$ have more than one ...
0
votes
1answer
45 views

Sums of two irreducible polynomials over $\mathbb{Z}$

Please help me to prove that any polynomial with integer coefficients can be represented as a sum of two irreducible polynomials over the ring $\mathbb{Z}$.
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2answers
45 views

Checking irreducibility of a polynomial over a finite field

A part of a coding theory course I am doing includes some questions on irreducible polynomials. I have a question with solution but am worried I have interpreted it incorrectly. So for $\mathbb F_5$ ...
1
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1answer
104 views

Let $R = \mathbb{Z} + x\mathbb{Q}[x]$. Find all the irreducibles in $R$.

Let $R = \mathbb{Z} + x\mathbb{Q}[x] \subset \mathbb{Q}[x]$. Find the irreducibles of $R$. Show that the irreducible elements in $R$ are $\pm p$ for prime integers $p$ and the irreducible ...
3
votes
1answer
60 views

Irreducible polynomial over a field $k$ with $char\ k = p > 0$

I'm studying for my Abstract Algebra II final and reviewing problems. I'm having some trouble with this one. Direction would be helpful. Let $k$ be a field with $char\ k = p > 0$, and let $f(x) ...
13
votes
9answers
358 views

Prove that $f=x^4-4x^2+16\in\mathbb{Q}$ is irreducible

Prove that $f=x^4-4x^2+16\in\mathbb{Q}[x]$ is irreducible. I am trying to prove it with Eisenstein's criterion but without success: for p=2, it divides -4 and the constant coefficient 16, don't ...
2
votes
2answers
71 views

How can I prove the polynomial f is irreducible

We have $f\in \mathbb{Z}_{3}\left[X\right],\:\:f=x^3+2x^2+a,\:\:a\in \mathbb{Z}_{3}$ and we need to find $a$ for which polynomial $f$ is irreducible. I looked on google but I don't understand very ...
2
votes
3answers
51 views

Irreducibility of polynomials over finite field of integers $\bmod 11$.

Theorem (Fermat's little Theorem). If $p$ is a prime and $a \in \mathbb{Z}$ with $a \nmid p$ then $a^{p-1} \equiv 1 \mod p.$ Let $\mathbb{Z}/p\mathbb{Z}$ denote the multiplicative group of ...
0
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0answers
28 views

Long division for multipolynomial expression, little o notation

I have this expression: $$\mathrm{Exp}=\frac{d^3(-12a^4)+d^2(4a^4-16a^3)+d(4a^3-6a^2-a)}{d^3(-12a^4+12a^3)+d^2(4a^4-20a^3+16a^2)+d(4a^3-11a+7a)+(1-2a+a^2)}$$ Is there any way I can take the second ...
2
votes
1answer
34 views

Show that $P = (f(x))$ is a maximal ideal of $F(x)$

Let $F$ be a field and $f(x)$ be an irreducible polynomial in $F[x]$. Prove that $P = (f(x))$ is maximal in $F[x]$. (Here is what I know: $f(x) \neq 0 \wedge f(x) \not \in U(F[x])$, since $f(x)$ is ...
8
votes
1answer
67 views

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

I want to know for what $n$, $$x^{2n}+x^n+1$$ is irreducible modulo 2. I think for $n=3^k$ but have no idea how to prove it.
0
votes
1answer
37 views

Minimal polynomial of $\sqrt[3]{2} + \omega$ over $\mathbb{Q}.$

Is the polynomial $f(x) = x^9 - 9x^6 - 27x^3 - 27$ irreducible over $\mathbb{Q}?$ I think it is because of Eisenstein's applied to the prime $3.$ Is it the minimal polynomial of $x = e^{2 \pi i/3} + ...
1
vote
1answer
24 views

“p-adic absolute value” in polynomial ring

I'm working with Gouvea's book on p-adic numbers. In problem 34 I'm asked to give a "p-adic" (I put it in quotes as in my understanding its just a p-adic-like) valuation and absolute value for an ...
0
votes
1answer
60 views

Show that $x^{8}+x^{4}+x^{3}+x+1$ is irreducible over $\mathbb{Z}_{2}[x]$

How do I show that $x^{8}+x^{4}+x^{3}+x+1$ is irreducible over $\mathbb{Z}_{2}[x]$? Someone says I should use the fact that the range of the matrix is 7, but I don't exactly know how that applies. ...