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

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2answers
14 views

Nilpotent elements of polynomial quotient ring

Let $F$ be a field and let $f\in F[x]$ be an irreducible polynomial. Are the non-units of $F[x]/(f^n)$ nilpotent elements?
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1answer
25 views

Irreducibility of a Polynomial after a substitution

I am trying to determine whether the polynomial $f(x) = x^6 + 34x^4 + 4x^2 + 89 \in \mathbb{Z}[x]$ is irreducible over $\mathbb{Z}$. Eisenstein's criterion doesn't help and I suppose I could determine ...
3
votes
2answers
54 views

Prove that $f(x)$ is irreducible iff its reciprocal polynomial $f^*(x)$ is irreducible.

This is what I'm trying to prove: Let $f(x)\in\mathbb{Q}[x]$ and $\deg(f(x))>1$. Prove that $f(x)$ is irreducible in $\mathbb{Q}[x]$ iff its reciprocal polynomial $f^*(x)$ is irreducible in ...
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votes
1answer
34 views

Show that a polynomial is still irreducible in a extension field

I have found this question on the Papantonopoulou's Algebra book: Let $f(x)$ and $g(x)$ be irreducible polynomials over a field $F$ with deg$ f(x) = 15$ and deg$ g(x) = 14$. Let $\alpha$ be a ...
1
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0answers
33 views

Irreducible variety

I have the following problem and seems to stuck with some basic understanding of irreducible and/or non-singular varieties. In $\mathbb{P}^3$ we have an irreducible variety $A$ given by two equations ...
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1answer
15 views

Let P(x,y,z) be an irreducible homogeneous second degree polynomial. Show that the intersection multiplicity of V(P) with any line l is at most 2.

I came across this question in Algebraic Geometry: A Problem Solving Approach: Let P(x,y,z) be an irreducible homogeneous second degree polynomial. Show that the intersection multiplicity of V(P) with ...
0
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1answer
36 views

Irreducible question

Can anyone show me how to prove that $y-x^3$ is irreducible in $\mathbb{A}^2(\mathbb{C})$ For my clarity, the questions asks to decompose variety $V(xy^4-x^7y^2) \subset \mathbb{A}^2(\mathbb{C}) $ ...
1
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1answer
53 views

How is $x^3 - x^2 - 1$ irreducible in $\mathbb{F}_3$?

My book says that this polynomial is irreducible over the said field. Clearly, the polynomial does not have root $r = 1$ since $(1)^3 -(1)^2 - 1 = -1 \not\equiv 0 \mod 3$. However, for $r = -1$, we ...
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votes
1answer
29 views

Irreductibility of $x^p+p-1$ in $\mathbb{Q}[x]$

Prove that the polynomial $x^p+p-1$ is irreducible in $\mathbb{Q}[x]$, with $p$ prime. I have been stuck on this problem for a while. Could anyone give me a hint?
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2answers
26 views

Unclear proof of irreducibility on $x^m+1$

I came across this problem and frankly, it's unclear looking at the solution. Prove $x^m+1$ irreducible in $\mathbb{Q}(x)$ if and only if $m=2^k$ for $k \in \mathbb{N}$ Well the solution's short ...
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0answers
59 views

Show that $1+X^4$ is reducible over $\mathbb Z_p$ for every prime $p$. [duplicate]

Show that $1+X^4$ is reducible over $\mathbb Z_p$ for every prime $p$. MY ATTEMPT==>I have used Fermat's theorem for this as $X^{p-1}≡1\bmod p$, then this can also be written in the form ...
0
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0answers
40 views

Irreducible over $\mathbb Q[x]$ but reducible over $\mathbb F_p$ for all $p$ [duplicate]

To solve this question I have almost finished the proof but I need a little detail to be rigorous. Let $K$ be the prime fields $\mathbb Q$ or $\mathbb F_p$. Prove that $$f(x)=x^4+1\in ...
0
votes
1answer
54 views

Find all the Zeros and their multiplicities of $f(x)=x^5 +4x^4 +4x^3 -x^2-4x +1$ over $\Bbb Z_5$.

Find all the Zeros and their multiplicities of $f(x)=x^5 +4x^4 +4x^3 -x^2-4x +1$ over $\Bbb Z_5$. Firstly,I've found the zeros of $f(x)$,just by simply substituting the elements of $\Bbb ...
0
votes
1answer
48 views

What are the roots of quintics?

I've been teaching myself a bit of Galois theory and from what I understand, arithmetic operations ranging from addition to taking roots are not enough to express all of the roots of a general ...
0
votes
2answers
46 views

Irreducible implies Separable in a Finite Field

Proposition 37 on page 549 of Abstract Algebra, 3rd Ed. by Dummit and Foote claims that irreducible implies separable over a finite field. Suppose $p(x)$ is irreducible over a finite field of ...
1
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1answer
24 views

Irreducibility of special cyclotomic polynomial.

I'm trying to show that $f(x)=1+x^p+x^{2p}+\dots +x^{p(p-1)}$ is irreducible over $\mathbb{Q}[X]$. I'm well aware that cyclotomic polynomials are irreducible, however the (many) proofs of this ...
5
votes
4answers
65 views

$R = \mathbb{Z}[\sqrt{-41}]$, show that 3 is irreducible but not prime in $R$

I'm asked to show that 3 is irreducible but not prime in $R = \mathbb{Z}[\sqrt{-41}]$. And if $R$ is a Euclidean domain. To show that it's not prime I have $(1 + \sqrt{-41})(1 - \sqrt{-41}) = 42 = ...
0
votes
1answer
45 views

$I=(f_1, \ldots, f_n)\subset k[x_1, \ldots, x_n]$ with $f_i\in k[x_i]$ irreducible polynomials

Let $A=k[x_1,\ldots, x_n]$ and $I=(f_1, \ldots, f_n)\subset A$ with $f_i\in k[x_i]$ irreducible polynomials. Is it true that $I$ is a maximal ideal in $A$? $I$ is a maximal ideal $\iff$ $1\in ...
-1
votes
1answer
50 views

Why is $\mathbb{Z}[\sqrt{-5}]$ an integral domain? [duplicate]

I could use some help with this. I know that $\mathbb{Z}[\sqrt{-5}]=\{a+b\sqrt{-5\} }|a,b\in\mathbb{Z}\}$. I then put $$0=(a+b\sqrt{-5})(c+d\sqrt{-5})=ac-5bd+(ad+bc)\sqrt{-5}$$ which leaves me with ...
0
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0answers
37 views

Exhibit a reducible polynomial of the form $x^p -x-c$ having no roots in a field of characteristic 0

Is it possible for a polynomial, $x^p -x-c$ where $p$ is prime, to be reducible in a field of characteristic $0$, yet have roots in that field? I know for a fact that the general form is true, ...
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1answer
62 views

If $x^p−x−c$ is irreducible in $F[x]$ then it has no root in the field.

The complete problem appears in Hungerford's Algebra. Let $c\in F$, where $F$ is a field of characteristic $p$ ($p$ prime). Then $x^p−x−c$ is irreducible in $F[x]$ if and only if $x^p−x−c$ has no ...
2
votes
1answer
41 views

Minimal polynomial for $\alpha=\sqrt{3-2\sqrt{2}}$ over $\mathbb{Q}$

Find the minimal polynomial for $\alpha=\sqrt{3-2\sqrt{2}}$ over $\mathbb{Q}$ $\alpha=\sqrt{3-2\sqrt{2}} \implies -\alpha^2-3=2\sqrt{2} \implies (\frac{-\alpha^2-3}{2})^2-2=0$ $\implies ...
1
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1answer
164 views

Reducibility of $x^2+1$ in $\mathbb{Z}_n[x]$ [closed]

I want to prove or disprove the statement: $x^2+1$ is reducible in $\mathbb{Z}_n[x]$ $\iff$ there exists $a$ such that $a^2=-1$ in $\mathbb{Z}_n$. How can I prove or disprove the proposition?
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2answers
20 views

Is $x^p-ax-b$ with $a,b\neq 0$ irreducible in a field with characteristic a prime p?

It's a part of a bigger problem I'm facing. Not only I don't know how to prove it, I don't know if it's true or false at all (so I have no idea what to try to prove and so I don't know where to ...
1
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0answers
16 views

General Techniques - Number sets

There are many problems involving, proving numbers are irrational or not an integer and so forth (e.g roots of polynomials, size of an angle) What are some general techniques/tricks that I can use in ...
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2answers
63 views

Is $x^4+x^2+1$ an irreducible polynomial over $\Bbb Z/2\Bbb Z$?

Is $x^4+x^2+1$ an irreducible polynomial over $\Bbb Z/2\Bbb Z$? According to this site the answer is no. But I can't find the factors. Can you?
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1answer
26 views

$f,g \in \mathbb Q[x]$ , $f(a)=g(b)=0$ ; $f$ is irreducible in $\mathbb Q(b)[x]$ iff $g$ is irreducible in $\mathbb Q(a)[x]$?

Let $a,b$ be complex roots of irreducible polynomials $f(x),g(x) \in \mathbb Q[x]$ . Let $F:=\mathbb Q(a) , K:=\mathbb Q(b)$ ; then is it true that $f(x)$ is irreducible in $K[x]$ if and only if ...
0
votes
2answers
45 views

Discriminant of Quintic (Galois theory)?

I am working on the irreducible polynomial $x^5-Npx+p=0$ where $p$ is a prime and $N\in\mathbb Z_+$. I need to calculate the discriminant of this to determine its Galois group, background here by ...
2
votes
3answers
109 views

$t^4+t^3+t^2+t+1 $ has no linear factors

I am trying to understand why the given polynomial has no linear factors over $\mathbb{Q}$. I am trying to do it using elementary methods (no eisenstein criterion etc to show it's irreducible). I ...
2
votes
2answers
42 views

Can we use Eisenstein's Irreducibility Criterion to show that $x^4+1$ is not reducible in Q?

As such: Let $a(x)=x^4+1\in\mathbb{Q}\left[x\right]$. Then choose any prime $p$. By Eisenstein's Criterion, we see that $p\nmid 1$, $p\mid 0$ (since all coefficients of intermediate terms are 0), and ...
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1answer
32 views

Irreducible polynomial - Find roots [duplicate]

I would like to show that polynomial $x^3-xy^2+y+1$ is irreducible over $\mathbb{Q}(x)[y]$. I thought I could find the roots in using $\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$, but I found that the two roots ...
0
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1answer
32 views

Normal closure $L$ of $K=\mathbb{Q}(\sqrt{10+3\sqrt{13}})$ and structure of $\mathrm{Gal}(L/\mathbb{Q})$

Find a normal closure $L$ of $K=\mathbb{Q}(\sqrt{10+3\sqrt{13}})$ and describe the structure of $\mathrm{Gal}(L/\mathbb{Q})$ My idea is that the normal closure is $L=\mathbb{Q}(\sqrt{10+3\sqrt{13}}, ...
2
votes
1answer
47 views

Find $\mathrm{Gal}(L, \mathbb{Q})$ and prove that $\sqrt[3]{3} \in L$

I am really struggling on these consecutive questions on Galois Theory: Question 1: Suppose $L$ is a normal closure of $\mathbb{Q}(\sqrt[3]{2}, \sqrt[3]{5}) $ over $\mathbb{Q}$. Find ...
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2answers
72 views

Show that the polynomial $x^3-xy^2+y+1$ is irreducible in $\mathbb{Q}[x,y]$

I know that $\mathbb{Q}[y][x]=\mathbb{Q}[y,x]$; thus, we can see this polynomial as a polynomial with variable $x$ and with coefficients in $\mathbb{Q}[y]$. Someone explains to me that we could use ...
0
votes
1answer
38 views

Is the converse of “Let $f\in\mathbb Z[x]$. If $f$ is reducible over $\mathbb Q$, then it is reducible over $\mathbb Z$” true?

In Gallian there is a theorem which states that "Let $f\in\mathbb Z[x]$. If $f$ is reducible over $\mathbb Q$, then it is reducible over $\mathbb Z$." This is an implication statement, i.e., its ...
3
votes
4answers
68 views

Find an irreducible polynomial of degree $21$ in $\mathbb{F}_2[x]$

Note: this is from a paper on Galois Theory, so I believe the technique will come from Galois... Possibly. I am going to reduce it to the monic case: $x^{21}+a_{20}x^{20}+... a_0$ where all $a_i$ ...
2
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1answer
33 views

Proving irreducibility; What is this method and what is the logic behind it?

The only two methods I know are Eisenstein's method Irreducibility modulo $n$ Now, I am asked the following question Show whether or not $p(x)=x^5-5x^4+10x^3-7x^2+8x-4$ is irreducible over ...
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votes
1answer
109 views

Prove that $X^6 - 10X^3 +8$ is irreducible over $\mathbb{Q}$. [closed]

Prove that $X^6 - 10X^3 +8$ is irreducible over $\mathbb{Q}$.
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1answer
32 views

Irreducible factorisation of polynomial over quotient field

Let $F=\mathbb{Z}_3[x]/<x^2+1>$. Factor $x^4+2$ into irreducibles in $F[x]$. I know that $F$ is a field since $x^2+1$ is irreducible. The usual way to find out that a polynomial is irreducible ...
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0answers
28 views

General Multiplication Algorithm for Finite Field of GF(p^m) for small values of p^m

I am working on a small computer program to generate multiplication table of general finite field in form of GF(p^m) where p is a prime number and m is an integer greater than one. While the special ...
1
vote
1answer
29 views

Ireducible polynomial over $\mathbb Z_4$

How can you prove that $f(x)=X^2+1$ is ireductible over $\mathbb Z_4$, the quotient ring? We know that $\mathbb Z_4$ admits divisors of $0$, as $2*2=0$, so any elemanary approach using $h\times g=f$ ...
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1answer
40 views

Show that the polynomial $P = x^8-6x^3+ 2x^2+2$ is irreductible in $\mathbb{Q}[x]$

Show that the polynomial $P = x^8-6x^3+ 2x^2+2$ is irreductible in $\mathbb{Q}[x]$. Is is possible to use Eisenstein's criteria? Otherwise, is anyone could help me to solve it?
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0answers
18 views

Decomposition of a polynom in a field extension

Let $K$ a field, $P \in K[X]$ irreducible of degree $n$, $L$ an extension field of $K$ with degree $m$ and $d=gcd(m,n)$. I want to show that for every $Q$ irreductible factor of $P \in L[X]$, ...
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votes
1answer
49 views

Polynomial - Irreducibility

Show that the polynomial $P = x^3-xy^2+y+1$ is irreducible in $\mathbb{Q}[x,y]$. I know that $(\mathbb{Q}[y][x]) \cong \mathbb{Q}[y,x]$, thus we can see $P$ as a polynomial with variable $x$ et ...
1
vote
1answer
40 views

Polynomial irreducibility and perfect square.

Hi guys I am trying to show the following statement and would appreciate if people can take a look and comment if I am on the right track. $g(x,y)=y^2-p(x)$ is irreducible if and only if $p(x)$ is ...
1
vote
1answer
31 views

Is $x^p+p-1$ always irreducible in Q[x] for p prime?

Is $x^p+p-1$ always irreducible in Q[x] for p prime? I have a feeling it is true, however im only able to prove it for p=2,3.How could i generalize it for every p? Thanks
4
votes
1answer
32 views

Is $t^4 + 7$ irreducible over $\mathbb{Z}_{17}$?

There aren't any linear factors, $-7$ doesn't have a square root in $\mathbb{Z}_{17}$ so I've ruled out those 'easy' factorisations. I'm not sure how to continue. Looking for a hint, not a solution.
1
vote
1answer
54 views

is f(x) = 0 irreducible in $\mathbb{Z} /2 \mathbb{Z}$?

Let's say I have a polynomial like $$f(x) = 4x^2 +12x +28$$ when I reduce this with respect to mod 2; I end up with $0$. Can I say that zero is irreducible in ...
1
vote
0answers
45 views

Is this polynomial irreducible over $\overline{\mathbb{Q}}$?

Let $d$ be a natural number greater than or equal to $3$. I have the following polynomial $$ F(x_1, ..., x_3) = x_1^d + x_2^d - x_3^d - ( x_1 +x_2 - x_3 )^d. $$ I am trying to figure out if this ...
1
vote
1answer
29 views

Separable polynomials are the product of the minimal polynomials of their roots?

I see the following claim in this answer: Since $f$ is separable, it follows that $f(x)$ must be the product of minimal polynomials of [its roots] But, I don't know how we justify this claim. ...