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

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Finding a cubic polynomial whose splitting field over $\mathbb{Q}$ equals $\mathbb{Q}(a)$ if $a$ is any of its roots

Question: Let $\alpha$, $\beta$ and $\gamma$ be the roots of a rational cubic polynomial $q$. Can we find a (non-trivial) example where the splitting field of $q$ over $\mathbb{Q}$ equals ...
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1answer
22 views

polynomials and minimality

Could someone explain the concept of minimal polynomials? It seems like these are polynomials which cant be reduced further, but at the same time I am confused cause when we consider $\mathbb Z_2[x]$ ...
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0answers
35 views

Irreducible polynomial

Does there exist an irreducible polynomial over a field K with two roots $a,b$ and $k\in K$ such that $a=b+k$ ? This can't happen if K is of characteristic $0$ , but can it happen if K is of ...
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0answers
49 views

Prove that $T^{4} -12T^{2} +64$ is irreducible in $\mathbb{Q}[T]$

Is the following correct? I choose $3$ irreducible in $\mathbb{Z}$. If $g=(1+(3))T^{4} - (12 + (3))T^{2} + (64 + (3)) \in \mathbb{Z[T]}/(3)$ is irreducible, then $f=T^{4} -12T^{2} +64 \in ...
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0answers
48 views

Polynomial of Degree 3 Solutions [duplicate]

If $p(x) \in F[x]$ is of degree $3$, and $p(x)=a_0+a_1x+a_2x^2+a_3x^3$, show that $p(x)$ is irreducible over $F$ if there is no element $r\in F$ such that $a_0+a_1r+a_2r^2+a_3r^3 =0$. If $p(x)$ is ...
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4answers
47 views

Reduction modulo p

I am going to begin the Tripos part III at Cambridge in October (after going to a different university for undergrad) and have been preparing by reading some part II lecture notes. Here is an extract ...
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1answer
56 views

Characterizing maximal ideals in $\mathbb{Z}[x]$

I need to prove this: Let $I\subset\mathbb{Z}$ be the ideal generated by $\{p,f(x)\}$, with $p$ prime in $\mathbb{Z}$. Then $I$ is maximal iff $f(x)$ is irreducible modulo $p$. So I was trying to ...
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1answer
25 views

If p(x) $\in F[x]$ is of degree 3, and $p(x) = a_0 + a_1*x + a_2*x^2 + a_3*x^3$.

If p(x) $\in F[x]$ is of degree 3, and $p(x) = a_0 + a_1*x + a_2*x^2 + a_3*x^3$, show that p(x) is irreducible oer F if there is no element $r \in F$ such that $a_0 + a_1*r + a_2*r^2 + a_3*r^3$. So ...
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0answers
7 views

Solutions to a polynomial equation in a PAC field not lying in a subfield

Suppose $f(x,y)$ is an absolutely irreducible polynomial over a PAC (pseudo algebraically closed) field $K$ such that $x,y$ actually appear in $f$. Let $L$ be a proper subfield of $K$. Are we ...
3
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1answer
70 views

$f(x)$ is irreducible but $f(x^n)$ is reducible

Does there exist an irreducible polynomial $f(x)\in \mathbb{Z}[x]$ with degree greater than one such that for each $n>1$, $f(x^n)$ is reducible?
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0answers
192 views

Families of Polynomials Irreducible in $\mathbb{Z}$ but reducible in $\mathbb{Z}/p\mathbb{Z}$ for all primes $p$.

I am wondering if there exist classification of polynomials that are irreducible in $ \mathbb{Z}$ but reducible $\pmod p$ for all primes $p$. I am aware that $\Phi_n$ has this property if ...
3
votes
1answer
52 views

Nice polynomial reducibility: $x^n+4$

Problem: Find all $n\in \mathbb{N}$ such that $f(x)=x^n+4$ is reducible in $\mathbb{Z}[x]$. It seems $n=4k$ is the only one (the factorization follows easily from Sophie Germain's identity in this ...
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2answers
51 views

Galois extension preserves irreducibility

Consider a Galois extension $K/F$. Let $f\in F[x]$ be irreducible of prime degree. If $f$ has no roots in $K$, prove that $f$ is irreducible in $K[x]$.
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2answers
82 views

Isn't $x^2+1 $ irreducible in $\mathbb Z$, then why is $\langle x^2+1 \rangle$ not a maximal ideal in $\mathbb Z[x]?$

Isn't $ x^2+1$ irreducible in $\mathbb Z$, then why is $\langle x^2+1 \rangle$ not a maximal ideal in $\mathbb Z[x]$? $ x^2+1$ cannot be broken down further non trivially in $\mathbb Z[x]$. hence, ...
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2answers
37 views

If $p$ is a prime prove that $x^{p-1} - x^{p-2} + x^{p-3}- \cdots -x+1$ is irreducible over $Q$

If $p$ is a prime prove that $x^{p-1} - x^{p-2} + x^{p-3}- \cdots -x+1$ is irreducible over $Q$. $1$st Attempt: $x^{p-1} - x^{p-2} + x^{p-3}- \cdots -x+1 $ $= x^{p-2}(x-1)+x^{p-4}(x-1)+ \cdots ...
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1answer
55 views

A Question about the Proof of Eisenstein's Irreducibity Criterion

Statement: Let $f(x) = a_n x^n + a_{n-1} x^{n-1}+ \cdots + a_0 \in \mathbb Z[x]$. If there is a prime $p$ such that $p \nmid a_n, p \mid a_{n-1}, \dots,p \mid a_0$ and $p^2 \nmid a_0 $, then $f(x)$ ...
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1answer
41 views

A question from the mod p irreducibility test's proof

Let $p$ be a prime an suppose that $f(x) \in \mathbb Z[x]$ with $\deg f(x) \geq 1$. Let $f_1(x)$ be the polynomial in $\mathbb Z_p[x]$ obtained from $f(x)$ by reducing all the coefficients of $f(x)$ ...
3
votes
1answer
40 views

Irreducibility of a polynomial over a field

I'm trying to show that the polynomial $f(x) = \frac{x^5}{32}-3x-2$ is irreducible over $\mathbb Q$. Obviously $f$ doesn't have a root over $\mathbb Q$ so I tried to use Gauss lemma for $32f$ and ...
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0answers
79 views

When do two integral superellipses have 'nice' intersections?

A recent question posed the nonlinear system \begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases} for real $(x,y)$ and asked for the sum $x+y$. As noted by commentary in the question, this regrettably ...
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2answers
98 views

Existence of Irreducible polynomials over Z of any given degree which do not satisfy the Eisenstein's Criterion

I just came across the following interesting question which has been once discussed: Existence of Irreducible polynomials over $\mathbb{Z}$ of any given degree I was wondering if we could find such ...
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2answers
125 views

Congruence modulo primes or in a polynomial ring over ${\rm GF}(2)$

Let $p, q$ be primes. Then the linear congruence $$ap \equiv r\pmod q$$ can be solved for $a\in\mathbb Z$ and will have a unique solution for each value of $r$ such that $0\leqslant a<q$. Am I ...
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4answers
189 views

Euclid's proof of the infinitude of primes to prove this question

I'm trying to prove that if $k$ is a field, then there are an infinite number of irreducible monic polynomials in $k[X]$. My attempt of solution is use almost the same strategy of the Euclid's proof ...
5
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1answer
64 views

Irreducibility of some multivariate polynomials

Consider the polynomials $xw-yz\in A[x,y,z,w]$ and $x^n+y^n+z^n\in A[x,y,z]$, where $A$ is a commutative ring. I am curious to know what conditions on $A$ (factorial ring, algebraically closed field, ...
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19 views

Using Descarte’s rule of signs to determine the number of positive roots.

Using the Descarte’s rule of signs to determine the number of positive roots. \begin{equation} f(q)=[(k_f+k_d+k_p*(1-q))(\lambda_b* \gamma ...
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Sets of Prime Numbers Generated By an Irreducible Monic Polynomial

Given a non-constant integral irreducible monic polynomial $f(x)$, the prime factors of its value at integers $x\in\mathbb{N}$ forms a set $\mathcal{P}(f)$. Is it possible that ...
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1answer
66 views

Wikipedia incorrect about irreducibile polynomials and roots

Wikipedia states : If an univariate polynomial $p$ has a root (in some field extension) which is also a root of an irreducible polynomial $q$, then $p$ is a multiple of $q$, and thus all ...
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1answer
43 views

Exists a binary primitive pentanomial of degree $n$, for every $n\ge5$?

Or, in other words, prove (or disprove) this conjecture: $\forall n\ge5,\exists(i,j,k),n>i>j>k>0,\text{ such that}$ $\;x^n+x^i+x^j+x^k+1\text{ is a primitive polynomial in }GF(2)$. Also: ...
4
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71 views

Irreducibility of some polynomial

Let $p(x) = (1+ \cdots +x^k)^2 + (1+ \cdots +x^k) + 1$, for some $k \geq 2$ fixed. I would like to know if $p(x)$ is irreducible in $\mathbb{Q}[x]$.
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Analogue of Fermat's primality test for polynomials and irreducibility

We've got Fermat's primality test to test if a number is probable prime. Is there an analogous test for polynomials in $\mathbb{F}_{p^n}[X]$ and irreducibility?
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84 views

Eisenstein's Criterion stronger version

So we all know about Eisenstein's Criterion and how useful it turns out to be. In particular, if we have a monic cubic $x^3+ax^2+bx+c$, it is irreducible if there exists a prime number $p$ such that ...
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112 views

On factorization of polynomials

I would be very grateful if you give me a hint on problem 9, section 3.6 of Hungerford Algebra, regarding factorization in polynomial rings, saying that: Suppose $f(x)= ...
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1answer
46 views

Number of monic irreducible polynomials over a finite field

Let $\mathbb{K}=\mathbb{F}_q$ and $\nu_n$ denote the number of monic irreducible polynomials over $\mathbb{K}$. It holds $$\nu_n=\frac{1}{n}\sum_{d\mid n}\mu\left(\frac{n}{d}\right)q^d$$ What I need ...
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2answers
61 views

Basic irreducible polynomial

I'm studying cyclic codes over a ring $R$. It is well known that a cyclic code over $R$ of length $n$ is an ideal of $R\left[ x \right]/\left( {{x^n} - 1} \right)$. Hence the factorization of ...
2
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1answer
84 views

What do we know about $\displaystyle \frac{f}{\gcd(f,f')}$ if $f\in\mathbb{F}_{p^d}[X]$?

Let $\mathbb{K}=\mathbb{F}_{p^d}$ and $f\in\mathbb{K}[X]$ be a non-constant polynomial with the factorization $$f=\prod_{i=1}^nf_i^{k_i}$$ where $f_i\in\mathbb{K}[X]$ is irreducible and ...
4
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2answers
70 views

Is $x^8 + x^5 + x^3 + x^2 + 1$ an irreducible polynomial or not in GF $2^8$

Is $x^8 + x^5 + x^3 + x^2 + 1$ an irreducible polynomial or not in Galois Field $2^8$? Thanks in advance.
3
votes
1answer
34 views

Existence of “simple” irreducible polynomial of degree 12 in a finite field

Assume that we have a finite field $\mathbb{F}_p$, where $p$ is prime, $p \equiv 1\ (\textrm{mod}\ 4)$ and $p \equiv 1\ (\textrm{mod}\ 3)$. I was looking for irreducible polynomial in a form $X^{12} + ...
4
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1answer
210 views

Proof that a Polynomial is Irreducible

In my abstract algebra class, we were given some polynomials to prove irreducible or not over the rationals. I have come up with a proof for this one, but I'm not sure if it's valid. The ...
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15 views

another irreducible polynomial

Let $\displaystyle f(x) = \sum_{i=0}^n a_ix^i$ be the integer polynomal such that $|a_i| \le N$ with $i=0,1,2,...,n$. Prove that if there exist $2N+4$ integers such that all of the absolute value of ...
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Again with an irreducibility

Prove that for any integer $d>3$, polynomial $\displaystyle p(x)=x^{d}+(-2d+1)x^{d-1}+\sum_{k=0}^{d-2}(k^2+k+1)x^{k}$ is irreducible. This is another problem from the sheet given to me by my ...
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0answers
29 views

Irreducible polynomial

Let $\displaystyle (b_0,b_1,b_2,b_3)$ be a permutation of the set $\displaystyle \{54,72,36,108\}$. Prove that $\displaystyle x^5+b_3x^3+b_2x^2+b_1x+b_0$ is irreducible in $\displaystyle \mathbb ...
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2answers
39 views

Polynom irreducibility over $\mathbb Z_5$

So, I have polynom over $\mathbb Z_5$: $x^8 - x^7 + 2x^6 + x^5 + 2x^4 + 2x^2 +3x +1$ and I have to find his irreducibile factors. How to do that? I can find his roots by replacing $x$ with $[1]_5, ...
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91 views

Showing that minimal polynomial has the same irreducible factors as characteristic polynomial

I'm trying to show that the minimal polynomial of a linear transformation $T:V \to V$ over some field $k$ has the same irreducible factors as the characteristic polynomial of $T$. So if $m = ...
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1answer
72 views

Showing that $x^n -2$ is irreducible in $\mathbb{Q}[X]$

I'm trying to show that the polynomial $X^n -2$ ($n \in \mathbb{N}$) is irreducible in $\mathbb{Q}[X]$ but am a bit stuck. Methods I know to show irreducibility: Gauss' lemma - which says that if I ...
3
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2answers
87 views

Normal if and only if is UFD

If we consider $f \in \mathbb{C}[x,y]$ an irreducible polynomial, then it is true that the domain $ \mathbb{C}[x,y]/(f)$ is normal iff it is UFD? I think this is false. I was trying to prove ...
2
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1answer
52 views

Characterisation of Galois Group with the action of $\sigma \in S_n$ on the roots

Let $f \in K[X]$ be irreducible and separable with roots $x_1,...,x_n$ in a splitting field $L$ of $f$ over $K$. We identify $\text{Gal}(L|K)$ with $\text{Gal}(L|K)\cong G\subset S_n$. How can I see ...
6
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1answer
54 views

Simple field extension and roots of a polynomial

Let $K$ be a field, $f \in K[X]$ separable and irreducible with $\text{deg}(f)=n$; $x_1,...,x_n$ are the roots of $f$ in a splitting field of $f$ over $K$. Let $g \in K[X]$ be any polynomial with ...
0
votes
1answer
20 views

$\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$?

I want to find out if this affermation is true: let $f\in \mathbb{Q}[x]$ such that $\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$? (We know that it has ...
8
votes
3answers
174 views

Polynomial irreducible - maximal ideal

I have a couple of ideals which I wonder if I correctly classify as maximal/prime ideal. $I_1 = \langle 2x^2 + 9x -3\rangle$, $I_2 = \langle x - 1\rangle$ $\mathbf 1)$ Is $I_1$ a maximal ideal in ...
0
votes
1answer
60 views

Why is $x^2-2ux+1$, where $u = \cos(\frac{2\pi}{n})$, irreducible in $\mathbb Q(u)$?

My textbook states that $x^2-2ux+1$, where $u = \cos(\frac{2\pi}{n})$ for some $n \in \mathbb N$ is clearly irreducible in $\mathbb Q(u)$. Is this obvious? I tried to write it as a product of ...
3
votes
3answers
64 views

$x^p-x+a$ irreducible for nonzero $a\in K$ a field of characteristic $p$ prime

Is it true that $f(x)=x^p-x+a\in K[x]$ is irreducible for nonzero $a\in K$ a field of characteristic $p$ prime? I've seen variants of this question around, but they don't seem to answer the question ...