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

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5
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2answers
733 views

Irreducible but not prime element

I am looking for a ring element which is irreducible but not prime. So necessarily the ring can't be a PID. My idea was to consider $R=K[x,y]$ and $x+y\in R$. This is irreducible because in any ...
2
votes
1answer
44 views

Is $4x^2+3xy^2+y^3+7$ irreducible in $\mathbb{C}[x,y]$?

Is $4x^2+3xy^2+y^3+7$ irreducible in $\mathbb{C}[x,y]$? I tried to group it like $(4)x^2+(3y^2)x+(y^3+7)$. This is a polynomial with degree $2$ so I am thinking of applying quadratic formula... where ...
7
votes
0answers
56 views

If $A[X] \cong B[X]$ as rings, are the degrees of irreducible polynomials the same in $A$ and in $B$?

First, I ask my question and then I add some explanations: Suppose that $A$ and $B$ are two commutative rings such that $A[X] \cong B[X]$ as rings. Denote by $D_A$ the set of all positive integers ...
5
votes
1answer
42 views

Does there always exist an irreducible polynomial of degree $d$ over $\mathbb{Z}/p\mathbb{Z}$? [duplicate]

Let $p$ be a prime and let $d$ be a positive integer. Does there always exist an irreducible (i.e. unfactorable) polynomial of degree $d$ over $\mathbb{Z}/p\mathbb{Z}$?
6
votes
3answers
77 views

Field with $125$ elements

I want to construct a field with $125$ elements. My idea is to consider the polynomial ring $\Bbb F_5[x]$. It is enough to find an irreducible polynomial $f\in \Bbb F_5[x]$ of degree $3$ because then $...
1
vote
1answer
28 views

Ring isomorphism $\Bbb Q[x]/(f)\cong \{c_0+c_1\alpha + c_2\alpha^2:c_i\in \Bbb R\}$

Let $f=x^3+x^2-2x-1\in \Bbb Q[x]$. Let $\alpha\in \Bbb R$ be a zero of $f$. $\Bbb Q[x]/(f)$ is isomorphic to the subring $R=\{c_0+c_1\alpha + c_2\alpha^2:c_i\in \Bbb R\}$ of $\Bbb R$. The map $\...
0
votes
0answers
24 views

Irreducible polynomial - Artin-Schreier exercise

Can somebody help me with this exercise? Is $f(x)=50x^6 + 6x^5 -10x^4 + 15x^3 + 4x -7$ irreducible in $\mathbb Z[x]$? I know I have to project in $\mathbb Z_5$ and it becomes an artin ...
2
votes
2answers
72 views

Number of irreducible polynomial over a field. [closed]

Find the number of irreducible monic polynomials of degree $2$ over a field with five elements. Please anyone help me.
3
votes
1answer
60 views

Is the polynomial $f(x) = x^4 + tx^3 + (t^2 + 1)x^2 + (t^3 + t)x + (t^4 + t^2)$ irreducible over $k(t)$?

Let $k$ be an algebraically closed field of characteristic 2 and let $k(t)$ be rational function field of one variable. Consider the polynomial $f(x) = x^4 + tx^3 + (t^2 + 1)x^2 + (t^3 + t)x + (t^4 + ...
4
votes
1answer
50 views

Is $x^{2\cdot 3^n}+x^{3^n}+1$ irreducible (mod 2)?

I'm new to the finite field theory, however after doing some trivial search on primitive polynomials, it seems that the polynomials of the form $$x^{2\cdot3^n}+x^{3^n}+1 \pmod 2$$ are irreducible. ...
2
votes
1answer
58 views

Solvability and reducibility of a polynomial in a “chain” of finite fields

This question is generalized based on my previous question: Is $x^5 + x^3 + 1$ irreducible in $\mathbb{F}_{32}$ and $\mathbb{F}_8$? Problem: Consider an irreducible polynomial $f = x^4 + x^3 + 1$ in ...
1
vote
2answers
81 views

Irreducible polynomial in finite-fields

Let $\overline{\mathbb{F}}_2$ be an algebraic closure of $\mathbb{F}_2$, and let $\alpha\in\overline{\mathbb{F}}_2$ be such that $\alpha^2+\alpha+1=0$. Prove that if a polynomial $P$ is irreducible in ...
3
votes
0answers
36 views

Methods of determining irreducibility of a polynomial in a large finite field

Given a finite field $\mathbb{F}_p$ and some polynomial $f(x)\in\mathbb{F}_p [x]$. What are some of the methods of determining the irreducibility of $f(x)$? I feel like there are many theorems that we ...
0
votes
0answers
44 views

Irreducibility of sums of two polynomials

I'm interested in a special type of polynomial factorization over $\mathbb {Q} $: testing the irreducibility of $f(x)+g(x)$, where $f$ and $g $ are relatively prime and $\text {deg}(f)<\text{deg}(g)...
3
votes
0answers
71 views

Can we continually factorize an expression like $x+y$?

I have a question that, for lack of familiarity or understanding of the relevant fields, I'm not quite sure how to formulate, so I'll just start off with an example and list some questions as I go. As ...
1
vote
1answer
55 views

Irreducible polynomials over $\mathbb{Z}_2[x]$

Prove that the polynomial $1+x+..+x^m$ is irreducible over $\mathbb{Z}_2$ if and only if $m+1$ is a prime number and 2 is a primitive root in $\mathbb{Z}_{m+1}$ Is there any proof without using ...
1
vote
0answers
88 views

Exact Probability of reducibility of Bivariate Polynomials

I am considering polynomials of the form $$P(x,y)= \sum_{k=0}^n\sum_{l=0}^n a_{k,l}x^{k}y^{l}$$ where $n \in \mathbb{N}$. The coefficients $a_{k,l}$ are considered to be randomly generated from the ...
3
votes
1answer
46 views

Is $x^3+y^3+z^3-1$ irreducible in a field $k$ of characteristic $\neq 3$?

If I can show that $y^3+z^3-1$ is irreducible, I believe I can use Eisenstein's criterion. But I don't see how to show even this. The other approach on my mind is to show that $k[x,y,z]/(x^3+y^3+z^3-...
11
votes
2answers
90 views

When is $X_1^{a_1} \cdots X_n^{a_n}-1$ irreducible?

Let $F$ be a field, and $a_1, ... , a_n \geq 1$ integers. When is the polynomial $$f = X_1^{a_1} \cdots X_n^{a_n}-1$$ irreducible in $F[X_1, ... ,X_n]$? I believe this should be the case if and ...
1
vote
2answers
105 views

Proof verification affine curve not isomorphic to plane curve

I'm trying to prove that the affine curve $X\subset\mathbb{A}^3$ given by $\alpha:\mathbb{A}^1\to\mathbb{A}^3$, $t\mapsto(t^3,t^4,t^5)$, is not isomorphic to a plane curve. Here is what I've done: it ...
16
votes
1answer
169 views

Is it possible for an irreducible polynomial with rational coefficients to have three zeros in an arithmetic progression?

Assume that $p(x)\in \Bbb{Q}[x]$ is irreducible of degree $n\ge3$. Is it possible that $p(x)$ has three distinct zeros $\alpha_1,\alpha_2,\alpha_3$ such that $\alpha_1-\alpha_2=\alpha_2-\alpha_3$? ...
1
vote
1answer
81 views

Prove that the polynomial $\prod\limits_{i=1}^n\,\left(x-a_i\right)-1$ is irreducible in $\mathbb{Z}[x]$.

Let $n>1$ be an integer. For $a_1,a_2,\ldots,a_n\in\mathbb{Z}$ with $a_1< a_2< a_3 < \dots < a_n$, prove that the polynomial $$f(x)=(x-a_1)(x-a_2)\cdots(x-a_n)-1\,.$$ is irreducible in ...
1
vote
1answer
87 views

Dependence of algebraic elements in a finite field

Lets work over the finite field $\mathbb{F}_p$ for a prime $p$. Consider a monic irreducible polynomial $f(X)=X^3+aX^2+bX+c$ in $\mathbb{F}_p[X]$. Let $x$ be a root of $f(x)=0$ (say, in the closure of ...
0
votes
1answer
40 views

Definition of Irreducible polynomial in terms of the unit of Integral domain.

Let $D$ be an integral domain. A polynomial $f(x)$ from $D[x]$ that is neither the zero polynomial nor a unit in $D[x]$ is said to be irreducible over $D$ if, whenever $f(x)$ is expressed as a product ...
2
votes
0answers
50 views

Factoring $x^5+B x^4+C x^3+D x^2+E x+F=(x^2+a x+b)(x^3+p x+q)$ over $\mathbb{Q}$

For a quntic polynomial to be reducible to the following form over $\mathbb{Q}$: $$x^5+B x^4+C x^3+D x^2+E x+F=(x^2+a x+b)(x^3+p x+q)$$ We need to match the coefficients ($a=B$ obviously, so we ...
3
votes
3answers
119 views

Brumer quintic polynomials - is there a general formula for the roots?

There exist a family of quintic polynomials, called Brumer's polynomials (or Kondo-Brumer), which have the form: $$x^5+(a-3)x^4+(-a+b+3)x^3+(a^2-a-1-2b)x^2+bx+a,~~~a,b \in \mathbb{Q}$$ According to ...
4
votes
2answers
96 views

'Strange' trigonometric roots of $x^5-4x^4+2x^3+5x^2-2x-1$ - could someone explain?

This quintic equation has $5$ real roots: $$x^5-4x^4+2x^3+5x^2-2x-1=0 \tag{1}$$ The roots are, from left to right: $$x_1=\frac{\cos \frac{19}{22} \pi}{\cos \frac{1}{22} \pi}$$ $$x_2=\frac{\cos \...
1
vote
2answers
62 views

A solvable quintic with the root $x=(\sqrt[5]{p}+\sqrt[5]{q})^5$ - what are the other roots?

I derived a two parameter quintic equation with the root: $$x=(\sqrt[5]{p}+\sqrt[5]{q})^5,~~~~~p,q \in \mathbb{Q}$$ $$\color{blue}{x^5}-5(p+q)\color{blue}{x^4}+5(2p^2-121pq+2q^2)\color{blue}{x^3}...
10
votes
2answers
138 views

Only five solvable quintic equations of the form $x^5+ax^2+b=0$? What are their solutions?

According to Wikipedia there is only five solvable quintic equations of the form $x^5+ax^2+b=0,~~a,b \in \mathbb{Q}$ (up to a scaling constant $s$). $$x^5-2s^3x^2-\frac{s^5}{5}=0 $$ $$ x^5-100s^3x^2-...
1
vote
2answers
57 views

$f(x+a)$ irreducibility means $f(x)$ irreducibility

Let $a~\in~\mathbb{Z}$ and let $f(x)~\in~\mathbb{Z}\left[x\right]$. Suppose that $f(x+a)$ is irreducible over $\mathbb{Z}$. Prove that $f(x)$ is irreducible over $\mathbb{Z}$. My idea is: $f(x)=u(x)*...
4
votes
2answers
48 views

Find roots of polynomial in a finite field

I need to build a field $L$ of 121 elements and find how many roots polynomial $g=x^9-1$ has in $L$. Then to find all these roots. So, $121=11^2$ this is power of prime. We can build finite field of ...
0
votes
1answer
38 views

Find the numbers at which the polynomial is irreducible over $\mathbb{Q}$

How can I find the integers $a$ at which the polynomial $f(x)$ is irreducible over the field $\mathbb{Q}$? Thank you! $$f(x) = 5x^4 - 6x^3 - ax^2 - 4x + 2$$
2
votes
3answers
61 views

Show that polynomial is irreducible over $\mathbb{Q}$

How I can prove that polynomial $f(x)$, where$$f(x) = x^4 + 3x^3 + 3x^2 - 5$$ is irreducible over $\mathbb{Q}$? Thank you
2
votes
2answers
39 views

Construction of field extension for $[E:\mathbb F_{11}]=3$

Let $\mathbb F_{11}\subset E$. Construct a field extension $E$ of $\Bbb{F}_{11}$ such that $[E:\mathbb F_{11}]=3$ Answer: Let $f(x)=x^3+1 $ be a polynomial in $\mathbb F_{11}[x]$ with $deg(f)=3$. ...
0
votes
2answers
46 views

Prove that $q(x)$ does not divide $p(x)$?

"Let $F$ be a field and suppose that $p(x),q(x) \in F[x]$ are the two polynomials $p(x) = x^5 - x^4 + x^3 - x^2 + x - 1$ and $q(x) = x^2-1$ (i) Prove that $q(x)$ does not divide $p(x)$ when $F = \...
4
votes
4answers
89 views

Show that $\mathbb{F}_9 \not \subset \mathbb{F}_{27}$

The usual answer will go like this: Since $2 \not | \ 3$ and $\mathbb{F}_{p^r} \subset \mathbb{F}_{p^s}$ if and only if $r | s$, then $\mathbb{F}_9 \not\subset \mathbb{F}_{27}$. However, I'm ...
2
votes
1answer
53 views

Possibilities for $\deg f$ if $\text{Gal}(f/\mathbb{Q})=Q_8, D_8$

Let $f$ be irreducible over $\mathbb{Q}$ with splitting field $F$. Suppose $\text{Gal}(F/\mathbb{Q})$ is either $D_8$ or $Q_8$. What are the possibilities for $\deg f$? I'm using Dummit & Foote,...
7
votes
1answer
89 views

Number of even irreducible monic polynomials of a given degree over a finite field

It is well-known that the number of irreducible monic polynomials of degree $n$ over the finite field of $q$ elements is given by the formula $$\frac{1}{n}\sum_{d\mid n}\mu\left(\frac{n}{d}\right)q^{d}...
-3
votes
1answer
38 views

Irreducible polynomial on $\mathbb{Z}_2$-field

I've found a theorem in the book "Linear Groups" (Dickson, 1901, p.16.): "In $\mathbb{Z}_2$, the degrees of the irreducible divisors of $x^{2^m}-x$ are divisors of $m$." I've read the prove in this ...
0
votes
1answer
30 views

If $a$ is algebraic and $f\colon\mathbb{Q}[x]\to\mathbb{C}$ where $f(g(x))=g(a)$, prove that $\ker(f)$ is a maximal ideal of $\mathbb{Q}[x]$

If $a$ is algebraic, then a polynomial $p(x)$ in $\ker(f)$ is irreducible iff it generates $ker(f)$. For an ideal $I$ in $Q[x]$ containing $\ker(f)$, let $p(x)=\ker(f)$ and $q(x)=I$. Then $p$ is ...
0
votes
3answers
39 views

If $a$ is algebraic, prove that there is a minimal polynomial $p(x)$ in $Q[x]$ such $p(a)$ = $0$.

If $f_a$: $Q[x]$ -> $C$ is the evaluation at $a$ map, then a polynomial $q(x)$ in $ker(f_a)$ is irreducible iff it generates $ker(f_a)$. Let $ker(f_a)$ = $h(x)$ so that $h(x)$ is irreducible and $f_a(...
4
votes
1answer
128 views

Minimal polynomial for $x=\tan \left( \frac{2}{5} \arctan p \right)+\tan \left( \frac{3}{5} \arctan p \right)$

I found numerically that the minimal polynomial for: $$x=\tan \left( \frac{2}{5} \arctan p \right)+\tan \left( \frac{3}{5} \arctan p \right)$$ has the following form: $$(3p^2-1)(p^2-1)\color{blue}...
3
votes
2answers
53 views

If $q(X)$ is reducible in $\mathbb Z[X]$, then it's reducible in $\mathbb Z_p[X]$ for every prime $p$

My book states, without a proof, that If $q(X)$ is reducible in $\mathbb Z[X]$, then it's reducible in $\mathbb Z_p[X]$ for every prime $p$. The contrapositive of the above result is more useful:...
0
votes
1answer
56 views

Irreducible over $\mathbb{Q}$(ring $\mathbb{Z}$)

How to proof irreducible over $\mathbb{Q}$(ring $\mathbb{Z}$) polynomials $f = (x-{a_1})\dots(x-{a_n})-1$ and $g = (x-{a_1})^2\dots(x-{a_n})^2+1$, {${a_i}$} - pairwise distinct integers.
2
votes
1answer
19 views

Ideal generated by two irreducible polynomials is the field itself

The question is: Let $F$ be a field and $f(x),g(x) \in F[x]$. Verify that $$N=\{r(x)\ f(x)+s(x)\ g(x):r(x),s(x)\in F[x]\}$$ is an ideal of $F[x]$. Then show that if $f(x)$ and $g(x)$ have different ...
1
vote
0answers
27 views

Irreducible polynomial of every degree over finite field

The existence of polynomials in title has been asked as a problem on MathStack many times; some answers were using existence of finite bigger fields, and some answers concern Mobius function with ...
1
vote
2answers
38 views

Reducibility of $x^q -x -1$ in $\mathbb{F}_{q}$

I came across the following excercise and do not know how to go about this. Given the polynomial $x^q -x -1$ in $\mathbb{F}_{q}$. Consider $q=8$. Show this polynomial is reducible by considering an ...
0
votes
1answer
32 views

For a field $K$, show that $f(x)=x^4+x^2+1\in K[x]$ is not a unit and not irreducible.

For a field $K$, show that $f(x)=x^4+x^2+1\in K[x]$ is not a unit and not irreducible. What I tried: To show that $f$ is not a unit I did the following. Suppose that $f$ is a unit, then there exists ...
2
votes
2answers
50 views

Question on splitting field and irreducible polynomials.

Let $K$ be a field, and consider a monic irreducible polynomial $f(x) \in K[x]$. Denote $d = \deg(f)$, and let $g(x) = f(x^2)$. Furthermore, let $\alpha \in \Omega^g_K$ (the splitting field of $g$ ...
1
vote
1answer
39 views

Why are these two calculations in $GF(2^5)$ not equal [closed]

I have a quick question about Galois fields, since there seems to be something I have misunderstood way back in university. Addition and subtraction in Galois fields are both done using XOR ...