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

learn more… | top users | synonyms

1
vote
2answers
47 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)*...
3
votes
2answers
38 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
37 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
56 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
35 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
42 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
82 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
51 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
76 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}...
0
votes
0answers
55 views

Examples of irreducible polynomials over a finite field with prescribed coefficients [closed]

I came to know that it is an open problem, but I am not able to find any simple example to explain it properly. Can some one help me with some simple explanation regarding what this problem is about?...
-3
votes
1answer
37 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
29 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
34 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
121 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
55 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.
-1
votes
0answers
38 views

irreducibility of monic polynomials over Z [closed]

Statement : Monic polynomials irreducible over Q are irreducible over Z. Where the polynomials belong to Z[x]. How to prove or disprove the statement. It seems like the converse of gauss lemma ...
2
votes
1answer
18 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
24 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
31 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
48 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 ...
0
votes
1answer
39 views

Is it possible to have a such polynomials?

An exercise asks me to write an example of such polynomials, if they exist: an irreducible polynomial of degree 5 in $\mathbb{R}[x]$. a polynomial of degree 5 in $\mathbb{R}[x]$ that has no roots a ...
-2
votes
1answer
48 views

Give two polynomials in $\mathbb Q[x]$ (of degree 2 and 3) such that their product is an irreducible polynomial in $\mathbb Q[x]$ of degree 5 [closed]

I know that $$x^k - p, \ \ \forall k>0\in N$$ is irreducible in $\mathbb Q[x]$ (Eisenstein theorem). I need two polynomials in $\mathbb Q[x]$ (one of degree 2 and another of degree 3) such that ...
0
votes
0answers
30 views

Prime factors polynomials

I have proved a theorem which I will state: For $f(x)=x^n+\sum_{i=1}^nh_ix^{n-i}$ a polynomial of degree $n$ where $h_i=r_i+d_i$ with $r_i$ real and $d_i$ infinitesimal. Then if $x^n+\sum_{i=1}^nr_ix^{...
0
votes
0answers
45 views

Show that $f(x)=x^3+(2+i)x+(1+i)$ is irreducible in $\mathbb{Z}[i][x]$

Problem says: Show that $f(x)=x^3+(2+i)x+(1+i)$ is irreducible in $\mathbb{Z}[i][x]$. I think I have to use Eisenstein's criterion with substituting $x$ with something but for me to show that ...
0
votes
1answer
17 views

Knowing the Galois group of the splitting field of a polynomial $f$, how can I show that $f$ is irreducible in the ground field?

So I'm given $f(x) = \sum_{k=0}^{8}\frac{x^k}{k!} \in \mathbb{Q}[x]$. Denote its splitting field by $E$, then I'm also given that ${\rm Gal}(E/\mathbb{Q}) \cong A_8$. The task is to prove that $f(x)$ ...
3
votes
1answer
88 views

Seeking an “easy ” way to show that $p(x)=x^6+\cdots+x^2+x+1$ is irreducible over $\Bbb{Z_{17}}$

As the title suggests, we need to Show that $p(x)=x^6+\cdots+x^2+x+1$ is irreducible over $\Bbb{Z_{17}}$ We can immediately answer that it is indeed irreducible since it is the cyclotomic ...
1
vote
1answer
38 views

Leading term ideal irreducible iff the ideal is irreducible?

Let $\mathbb{K}$ be a field. Given an ideal $I \subset \mathbb{K}[x_1,\dots, x_n]$ and a monomial order we can consider the ideal $LT(I) = (lt(f) \ | \ f\in I )$, where $lt(f)$ denotes the leading ...
1
vote
1answer
55 views

Polynomial that is irreducible over $ \mathbb{Q} $ but reducible over every finite field [duplicate]

I want to prove that $ X^4 - 10X^2 + 1 $ is reducible in $ \mathbb{F}_p[X] $ for every prime number $ p $, but it is irreducible over $ \mathbb{Q} $. I am not sure how to approach this problem; any ...
2
votes
4answers
180 views

Prove that $X^4+X^3+X^2+X+1$ is irreducible in $\mathbb{Q}[X]$, but that it has two different irreducible factors in $\mathbb{R}[X]$

Prove that $X^4+X^3+X^2+X+1$ is irreducible in $\mathbb{Q}[X]$, but it has two different irreducible factors in $\mathbb{R}[X]$. I've tried to use the cyclotomic polynomial as: $$X^5-1=(X-1)(X^4+X^3+...
0
votes
1answer
18 views

let $P$ $\in$ $\mathbb{F}_{p}[x,y]$, if $P$ does not have multiple factors then $P-1$ is irreducible. [closed]

I need to show the following: Let $P$ $\in \mathbb{F}_{p}[x,y]$. If $P$ does not have multiple factors, then $P-1$ is irreducible." Please help.
0
votes
1answer
57 views

Proof that $a^{n}+b^{n}$ is irreducible over $\mathbb Q$

The sum of fourth powers cannot be factored over $\mathbb Q$, since $ a^4+b^4 = (a^2+\sqrt{2}ab+b^2)(a^2-\sqrt{2}ab+b^2)$ And these quadratic factors does not have any real rational factors. How ...
2
votes
1answer
32 views

what does $\alpha$ signify in finite fields modular arithmetic

Say $\frac{\mathbb{Z}_{2}\left [ x \right ]}{x^{2}+x+1}=\left \{0,1,\alpha ,1+\alpha \right \}$ is a finite field with its elements listed. I am finding it difficult to understand what it means ...
2
votes
2answers
303 views

How To Prove That The Rijndael Polynomial Is Irreducible?

I am learning about the AES algorithm which uses the finite field ${\mathbb{Z}_2[x]}\over{(p(x))}$, where $p(x)=x^8+x^4+x^3+x+1$. How do you prove that this polynomial is irreducible? I know that for ...
1
vote
0answers
20 views

Using the discriminant to find Galois group of a quartic

I am working on finding the Galois groups of polynomials - in particular polynomials of degree $4$ I know that if we have a polynomial of the form $X^4+pX^2+qX+r$ we can find its cubic resolvent. ...
3
votes
2answers
63 views

Galois groups of $x^3-3x+1$ and $(x^3-2)(x^2+3)$ over $\mathbb{Q}$

I want to find the Galois groups of the following polynomials over $\mathbb{Q}$. The specific problems I am having is finding the roots of the first polynomial and dealing with a degree $6$ polynomial....
1
vote
1answer
18 views

Irreducible polynomial of degree $5$ in $L[X]$ using Kummer's theory

Let $L=\mathbb{F_2}(\theta)$ where $\theta$ is a root of $X^4+X+1$ I am trying to solve the following consecutive questions in Galois Theory: Prove that $L$ has degree $4$ over $\mathbb{F_2}$ ...
1
vote
2answers
51 views

Is $X^5+…+1 \in \mathbb{F_2}[X]$ irreducible?

I am trying to determine if the following polynomials are irreducible in $\mathbb{F_2}[X]$ are irreducible: $f(X)=X^5+X^2+1$ $g(X)=X^5+X^3+1$ There are no linear factors since $f(0)=f(1)=g(...
2
votes
2answers
29 views

Irreducibility of $X^5-7$ over $\mathbb{Q}(\sqrt[7]{2})[X]$ and degree of spitting field

I have worked through these two questions but am unsure if I got the right idea, please may you help me? Prove that $X^5-7$ is irreducible over $\mathbb{Q}(\sqrt[7]{2})[X]$ Can we say that $f(X)...
1
vote
2answers
35 views

Irreducibility Over $\def\Q{\Bbb Q}\Q[\sqrt2]$

Prove that the $q(x)=X^2+3$ is irreducible over $\Q[\sqrt{2}]$. Is my proof correct? Proof. Since it is a polynomial of second degree it factors iff it has $2$ polynomials of degree $1$. Since we ...
0
votes
1answer
21 views

Given $F \subset L \subset K$ where $K$ is a Galois ext. of $F$, find an example where $F \subset L$ is not a Galois ext.

I have already shown that if $F\subset K$ is a Galois extension, then for any intermediate field $L$, we have $L\subset K$ is a Galois extension. I then want to show that it's not necessarily true ...
3
votes
1answer
28 views

Is there something to say about the irreducibility of polynomials and their derivatives?

Is there a relation between the irreducibility of a polynomial and its derivative under certain conditions?
1
vote
3answers
73 views

Minimal polynomial of $\sqrt[3]{2} + \sqrt{3}$

Suppose I want to find the minimal polynomial of the number $\sqrt[3]{2} + \sqrt{3}$. Now that means I want to find a unique polynomial that is irreducible over $\Bbb Q$ such that $f(x)=0$. Now I ...
1
vote
0answers
17 views

Moving a polynomial coefficient so it's irreducible

Given a polynomial over $Z$, $P(X)$, I'd like to show that for any $i$ in $N$, there is a whole number $r$ so that $P(x)+r*x^i$ is irreducible. I'd preferably would love a elementary argument so ...
3
votes
0answers
46 views

There is no irreducible polynomial of largest degree in $\mathbf{F}_q[x]$

I am asked to prove or disprove that given a finite field $\mathbf{F}_q$, the ring $\mathbf{F}_q[x]$ contains irreducible polynomials of arbitrarily large degree. I couldn't think of a reason why this ...
2
votes
2answers
49 views

Irreducibility of polynomials in $\mathbf{Z}_p[x]$ - understanding proofs

I am reading through some irreducibility proofs and there's something I don't quite understand: $x^3+2x+1$ is irreducible in $\mathbf{Z}_3[x]:$ no roots in $\mathbf{Z}_3$ and degree $3$ so ...