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

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Efficient Computation of Swinnerton Dyer Polynomials

the Swinnerton-Dyer polynomials are defined as $$SD_n(x) = \prod(x \pm \sqrt{2} \pm \sqrt{3} \pm ... \pm \sqrt{p_n})$$ where the product is taken over all possible permutations of $+$ or $-$ signs. ...
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58 views

Irreducible Polynomial in $\mathbb{Q}[x]$ [on hold]

Show that $x^4+2x^2+4$ is irreducible in $\mathbb{Q}[x]$.
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41 views

Why is 105th cyclotomic polynomial interesting?

According to wikipedia the 105th cyclomatic polynomial is interesting because 105 is the lowest integer that is the product of three distinct odd prime numbers and this polynomial is the first ...
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47 views

Irreducibility of $x^4+x^3+x^2+x+1$

How can I see that the polynomial $x^4+x^3+x^2+x+1$ is irreducible over $\mathbb Q$? I can't apply eisenstein's theorem. What are the other possibilities?
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1answer
73 views

Irreducible polynomials over the reals

Everybody knows that the degree of irreducible polynomials over the reals is either one or two. Is it possible to prove it without using complex numbers? Or without using fundamental theorem of ...
3
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1answer
52 views

A walkthrough of how to apply Eisenstein's criteria to show that a multivariate polynomial is irreductible.

I have found a few resources such as 2.0.3 here. There however are a series of "identifications" such as saying $k[x,y,z] = k[y,z][x]$ Why can I make these identifications? can I ALWAYS do so? I ...
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2answers
77 views

Elementary proof of the irreducibility of $T^4 - a T - 1$ in $\mathbf{Q}[T]$ when $a\in\mathbf{Z}-\{0\}$

This is from the exercises of Bourbaki, Algèbre, Chapitre V, first exercise of the exercises concerning the second paragraph of the fifth chapter. (p. 140.) As Gauss Lemma ("if your gcd is ...
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1answer
58 views

Proof $27X^3 - 13X^2 + 180 \in \mathbb{Q}[x]$ is irreducible without Gauss's lemma

I'm asked to show that $27X^3 - 13X^2 + 180 \in \mathbb{Q}[x]$ is irreducible in $\mathbb{F}_{13}[X], \mathbb{Z}[X]$ and $\mathbb{Q}[X]$. I've managed to proof the first two. In $\mathbb{F}_{13}[X]$ I ...
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1answer
30 views

Irreducible polynomial with three real zeros

Is there an irreducible (cubic) polynomial with rational coefficients with three real zeros? (When I speak of irreducibility I mean over rational numbers.) How about an irreducible polynomial of ...
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2answers
34 views

How to factorize polynomial in GF(2)?

I want to know how to factorize polynomials in $\ GF(2) $ without a calculator in a product of irreducible factors. For example, in my exercises, I must factorize $\ p(x) = (x^7 - 1)$ The response is ...
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1answer
47 views

How do I prove the following theorem?

If $F$ is a field and $f(x) \in F[x]$, then $f(x)$ has no repeated roots if and only if $(f, f') = 1$, where $f'$ denotes the derivative of $f$.
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56 views

Factorize $x^4+n(2-n)x^2+n^2$ into irreducible polynomials over Q for any natural number n.

Factorize $x^4+n(2-n)x^2+n^2$ into irreducible polynomial over $\mathbb{Q}$ for any natural number n. This is an extended version of my previous question here. I know if $n=4$, ...
2
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2answers
269 views

Back to Front Eisenstein - number theory

So I was reading through Springer's Elements of Algebra and it brought up the existence of a 'back to front' version of the criterion. It goes something like; Let $f(x) = a(n)x^n +a(n−1)x^{n-1} ...
2
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73 views

Show that $x^4+n(2-n)x^2+n^2$ reducible over Q for any natural number n.

Show that $$x^4+n(2-n)x^2+n^2$$ reducible over Q for any natural number n. Here is what I did $$x^4+n(2-n)x^2+n^2=x^4+2nx^2-n^2x^2+n^2=x^4+2nx^2+n^2-n^2x^2$$ $$=(x^2+n)^2-(nx)^2$$ ...
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1answer
44 views

An integer square matrix of prime order has size at least $(p-1)\times (p-1)$

There's$\let\geq\geqslant\DeclareMathOperator{\GL}{GL}$ this exercise in my algebra course book: Let $p$ be a prime and $A\neq I$ an $n\times n$ matrix over $\mathbb Z$ such that $A^p=I$. Prove ...
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38 views

If an irreducible $f(x)$ divides $f(x^2)$, then its splitting field is $F(\alpha)$ for any root $\alpha$ of $f(x)$?

Let $F$ be a field and let $f(x) \in F[x]$ be an irreducible polynomial. Suppose $E$ is an extension of $F$ which contains a root $\alpha$ of $f(x)$ such that $f(\alpha^2)=0$. Show that $f(x)$ splits ...
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32 views

Root finding using Galois theory

Is there a method in Galois theory that, say given an $n$th degree polynomial with integer coefficients $$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$$ and $\alpha_1$ is a root of said polynomial, gives ...
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1answer
18 views

possibility of trisection of angles

i know that $\frac{\pi}{7}$ can be trisected if and only if $4x^3-3x-cos( \frac{\pi}{7})$ is reducible over $\mathbb {Q}$$(cos\frac{\pi}{7}) $. but i don't know how to check this. help pls. thanks ...
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1answer
21 views

Show T being prime element in $ F_{2}(T) $

Show that $X^4+TX^2+T$ is irreducible in $ F_{2}(T) $ Using Eisenstein with T as a prime element this proof is simple. Can I proof that T is prime any easier than in the folowing: Theorem 1: K is ...
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1answer
53 views

Degree of field extension $F_{p}(X^p,Y^p) \subset F_{p}(X,Y) $

What is the degree of the following field extension? $$ K=F_{p}(X^p,Y^p) \subset F_{p}(X,Y)=L $$ Because of $ F_{p}(X^p,Y^p) \subset_{(1)} F_{p}(X,Y^p) \subset_{(2)} F_{p}(X,Y) $ and $(1)$ the ...
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Why can't this happen in fields $K$ with $\operatorname{char}(K)=0$?

Let $f(x) \in K[x]$ irreducible. Then: $f(x)$ separable $\Leftrightarrow $ $(f, f')=1$ $f$ is irreducible so $$ (f,f')=\left\{\begin{matrix} f\\ 1 \end{matrix}\right. $$ $f(x)=a_0+a_1 x + \dots + ...
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31 views

Polynomial for each extension degree

Let $E$ the splitting field of a polynomial in $\mathbb{Q}[x]$ of degree $3$, then $[E:\mathbb{Q}]=1,2,3,6$. I am asked to give an example for each case... Are the following correct?? ...
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1answer
36 views

Irreducibility of multivariate polynomials over algebraic numbers [duplicate]

Why a polynomial in $\bar{\mathbb{Q}}[x_1,\dots,x_n]$ is irreducible iff it's irreducible in $\mathbb{C}[x_1,\dots,x_n]$?
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1answer
45 views

Is there a faster way to factor $X^{12}-1$ over $\mathbb{F}_5[X]$?

Problem: Factor $X^{12}-1$ into irreducibles in $\mathbb{F}_5[X]$. This problem appeared on a past qual and took me awhile to do. While I solved it, I'll need to be able to do problems like this a ...
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1answer
42 views

How to conclude that whether a given polynomial is irreducible or not?

For which $n\in \{2,3,7\}$ is the polynomial $x^3+x^2+x+2$ irreducible in $\mathbb{Z}/(n)[x]$? My work: For a given ring $R$ and a polynomial $p(x)\in R[x]$, if $p(\alpha)=0$ for some $\alpha\in ...
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1answer
55 views

Find all the maximal ideals in the ring $\mathbb{R}[x]$.

Find all the maximal ideals in the ring $\mathbb{R}[x]$. My work: If $M$ is maximal, then $\mathbb{R}[x]/M$ is a field. Then if $M$ is of the form $(p(x))$ then $p(x)$ must be irreducible. So ...
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49 views

Show that $Y^2 + X^2(X+1)^2$ is irreducible over $\mathbf R$

Show that $Y^2 + X^2(X+1)^2$ is irreducible over $\mathbf R$. Are there some general tricks for avoiding barbaric computations in general case?
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1answer
68 views

Polynomial with a root modulo every prime but not in $\mathbb{Q}$. [duplicate]

I recently came across the following fact from this list of counterexamples: There are no polynomials of degree $< 5$ that have a root modulo every prime but no root in $\mathbb{Q}$. ...
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1answer
28 views

Find the minimal irreducible polynomial $Irr(a, \mathbb{Q})$

Let the field extension $\mathbb{Q} \leq \mathbb{C}$ and $a=e^{\frac{2 \pi i}{8}}$. I have to find the minimal irreducible polynomial $Irr(a, \mathbb{Q})$. $a$ is a root of the polynomial ...
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66 views

How to prove the polynomial $ x^{2011}-x\in \mathbb{Z}_{2011}[x] $ is separable?

I am trying to prove that the polynomial $ x^{2011}-x\in \mathbb{Z}_{2011}[x] $ is separable. Definition: Let K be a field and $f\in K[x]$ an irreducible polynomial. The polynomial $f$ is said to be ...
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Computer program for factorization into irreducible polynomials over $\mathbb{Z}_p^k$

Hensel's Lemma allows us to factor a polynomial uniquely into basic irreducible factors over $\mathbb{Z}_p^k$. Is there a SAGE or Magma command that gives this factorization? Or can anyone help in ...
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81 views

How to show easily that $X^4+8$ is irreducible?

Is there an easy way to show that $X^4+8$ is irreducible ? I was thinking aboute finding a $a$ such that I can use the Eisenstein criterion $(X+a)^4+8$, but I don't find a such $a$.
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73 views

Factor $X^7-(4+i)\in\mathbb{Q}(i)[X]$…if possible.

I think $X^7-(4+i)\in\mathbb{Q}(i)[X]$ is irreducible (simply because I don't know how to go about factoring it). Would it suffice to show that it is irreducible over $\mathbb{Z}[i]$? If so, I ...
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70 views

prove that $f\left( x \right) = x^3 + 3x - 1$ is irreducible in $Q\left[ X \right]$

prove that $f\left( x \right) = x^3 + 3x - 1$ is irreducible in $Q\left[ X \right]$ Let $\theta$ be a root of $f(x)$ compute $\frac{1}{\theta }$; $ \left( {2 + \theta ^2 } \right)^{ - 1} $ and in ...
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0answers
48 views

Irreducible polynomial and primes [duplicate]

Let $n$ be a prime number. How can I show that the polynomial $f(x) = x^{n-1} + x^{n-2} + x^{n-3}+ \cdots + x+ 1$ is irreducible over any finite field?
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1answer
42 views

Prove that the following polynomials are irreducible or not.

I want to show that: 1) $X^4+1$ is irreducible The roots are the elements of $$\left\{\frac{\pm 1+i}{\sqrt 2},\frac{\pm 1-i}{\sqrt 2}\right\}$$ therefore it's not a product of a polynomial ...
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2answers
78 views

How to factor polynomials in $\mathbb{Z}_n$?

How to factor a certain polynomial over $Zn$. for example factor the following polynomial into irreducible polynomials in $Z5$: $X^3+X^2+X-1$ or factor the following polynomial into irreducible ...
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2answers
50 views

Show that polynomial is reducible

Show that $p(x)$ = $x^3 + 3x^2 + 2x + 4$ is reducible in $\mathbb{Z}$$/$$7$$\mathbb{Z}$ Is the approach for this to factor it and then find a root? I'm a little confused on how to start. Any tips ...
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39 views

Getting the multiplicative inverse of a polynomial

I have a polynomial $m(x)= x^2 + x + 2$ that's irreducible over $F=\mathbb{Z}/3\mathbb{Z}$. I need to calculate the multiplicative inverse of the polynomial $2x+1$ in $F/(m(x))$. I'd normally use ...
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1answer
50 views

Irreducible in $\Bbb Q[x]$

Suppose $f(x)$ is an polynomial of integer coefficients. If for infinitely many integers $x$, $f(x)$ is prime. Show that $f(x)$ is irreducible in $\Bbb Q[x]$. Suppose $f(x)$ is is reducible in $\Bbb ...
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A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...
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Sufficient condition for a polynomial to split

I found a problem that reads: let $K\subseteq L$ be two fields, and consider an irreducible $f(x)$ in $K[x]$. Show that if there exists an $a\in L$ such that $a$ and $a^2$ are roots of $f$, then $f$ ...
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77 views

Show that $\mathbb Q(\sqrt p) \not\simeq\mathbb Q(\sqrt q)$

I'd like to show that for $p,q$ distinct primes, the extensions $\mathbb Q(\sqrt p),\mathbb Q(\sqrt q)$ are not isomorphic. I don't really have knowledge of the "high-level language" of algebraic ...
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Irreducibility of $~\frac{x^{6k+2}-x+1}{x^2-x+1}~$ over $\mathbb Q[x]$

The Artin—Schreier polynomial $~x^n-x+1~$ is always irreducible over $\mathbb Q[x]$, unless $n=6k+2$, in which case it seems to have only two factors, one of which is always $x^2-x+1$. The ...
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1answer
46 views

NTRU cryptosystem

In the NTRU cryptosystem we are dealing with convolution polynomial rings and we compute $f(x)= T(d+1,d)$ and $g(x)= T(d,d)$ but when calculating their inverse in $R_q=(Z/qZ[x] / (x^N-1))$ and ...
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57 views

Prove particular quintic is irreducible

The problem is to prove that the quintic $$x^5+10x^4+15x^3+15x^2-10x+1$$ is irreducible in the rationals. I don't have much knowledge in group theory, and certainly not in Galois theory, and I'm ...
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47 views

Factorization and Roots of the following Polynomials

I'm struggling with this exercise $(a)$$f:=T^4 +6T^2 -8T - 3 \in \mathbb{Q}[T]$. Show that f is irreducible, with exactly $2$ real roots. $(b)$ Let $\alpha$ und $\beta$ be two real roots of ...
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1answer
28 views

Show that this polynomial is irreducible over Q

I have to show that the polynomial $1+x^p+x^{2p}+...x^{p^2-p}$ where p is a prime is irreducible over rationals. I am only looking for a hint. How should I go about this?
3
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2answers
88 views

Why is $T^4+6T^2-8T-3\in\mathbb Q[T]$ is irreducible?

Why is $T^4+6T^2-8T-3\in\mathbb Q[T]$ is irreducible ? Obviously Eisenstein doesn't work, but can we do a substitution and obtain a form such that the criterion is applicable ? We didn't cover ...
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1answer
48 views

Is $\mathbb{Q}(\pi)$ a simple extension of $\mathbb{Q}\left(\frac{\pi^3}{1+\pi}\right)$?

In the case of an algebraic extension, I could think easier than this case. But I got stuck in this problem. I know that the dimension of $\mathbb{Q}(\pi)$ over ...