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

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2
votes
1answer
22 views

Show that $P = (f(x))$ is a maximal ideal of $F(x)$

Let $F$ be a field and $f(x)$ be an irreducible polynomial in $F[x]$. Prove that $P = (f(x))$ is maximal in $F[x]$. (Here is what I know: $f(x) \neq 0 \wedge f(x) \not \in U(F[x])$, since $f(x)$ is ...
6
votes
1answer
46 views

Irreducibility of $x^{2n}+x^n+1$

I want to know for what $n$, $$x^{2n}+x^n+1$$ is irreducible modulo 2. I think for $n=3^k$ but have no idea how to prove it.
0
votes
1answer
27 views

Minimal polynomial of $\sqrt[3]{2} + \omega$ over $\mathbb{Q}.$

Is the polynomial $f(x) = x^9 - 9x^6 - 27x^3 - 27$ irreducible over $\mathbb{Q}?$ I think it is because of Eisenstein's applied to the prime $3.$ Is it the minimal polynomial of $x = e^{2 \pi i/3} + ...
1
vote
1answer
21 views

“p-adic absolute value” in polynomial ring

I'm working with Gouvea's book on p-adic numbers. In problem 34 I'm asked to give a "p-adic" (I put it in quotes as in my understanding its just a p-adic-like) valuation and absolute value for an ...
0
votes
1answer
53 views

Show that $x^{8}+x^{4}+x^{3}+x+1$ is irreducible over $\mathbb{Z}_{2}[x]$

How do I show that $x^{8}+x^{4}+x^{3}+x+1$ is irreducible over $\mathbb{Z}_{2}[x]$? Someone says I should use the fact that the range of the matrix is 7, but I don't exactly know how that applies. ...
2
votes
1answer
25 views

How do we determine the decomposition of $p\mathcal{O}_K$ in $K = \mathbb{Q}(\sqrt[3]{5})$?

Let $K = \mathbb{Q}(\sqrt[3]{5})$, and $\mathcal{O}_K$ be its ring of integers. In general, how do we decide the decomposition of $p\mathcal{O}_K$, for an odd prime $p$? I know that by Kummer's ...
2
votes
1answer
21 views

Power of a root in a field

Suppose $p(x)$ is an irreducible polynomial over a field $F$. Let $\alpha$ be a root. Compute the powers of $\alpha$ in $F(\alpha)$. I am not sure what the powers of a root are and how to compute ...
2
votes
1answer
46 views

Techniques to prove that two field extensions are distinct

I have been trying to create a family of pairwise distinct field extensions from a one-parameter family of irreducible polynomials, but have no idea how to prove that they are distinct. One pair is $f ...
0
votes
1answer
16 views

Show that $\bar{a}_{n}(\bar{x})^n+···+\bar{a}_{1}\bar{x}+\bar{a}_{0}=0_{F[x]/I}$

Let $F$ be a field, $f(x)$ be an irreducible polynomial in $F[x]$ and $I =(f(x))$. Let $f(x)= a_nx^n+···+a_1x+a_0, a_i \in F$ for $i=0,...,n$. And, $\bar{x} = x + I ∈ F[x]/I$ and $\bar{a_i} = a_i + I ...
0
votes
1answer
12 views

Polynomial with degree less than degree of an irreducible polynomial of the same root is 0

Let $F$ be a field, and $p(x)\in F[x]$ be an irreducible polynomial. Suppose $\alpha$ is a root of $p(x)$. Show that if $q(x)$ is a polynomial such that $\deg q(x) < \deg p(x)$ and $q(\alpha)=0$, ...
1
vote
2answers
35 views

Example such that $ f_1(x)$ is reducible but $f(x)$ is irreducible.

(The (mod p) Irreducibility Test) 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 ...
0
votes
0answers
31 views

Degree $5$ Irreducible Polynomial Solvable by Radicals and Abelian Extension

Consider an irreducible polynomial of degree $5$ over $\mathbb{Q}$ which is solvable by radicals. How do I show that its splitting field is contained in a field of the form $K(a)$, where $K$ is an ...
2
votes
2answers
47 views

Why $\mathbb{Q} [x]/(x^{4}+1) \simeq \mathbb{Q}(i,\sqrt{2})$?

I do not understand why $\mathbb{Q} [x]/(x^{4}+1) \simeq \mathbb{Q}(i,\sqrt{2})$. I know that $x^{4}+1$ is irreducible $(f(x + 1) = (x + 1)^4 + 1$ is Eisenstein at $2$) and it has the roots: ...
1
vote
1answer
30 views

Galois group of $f$ is cyclic if $\deg f$ is prime

Hello I am learning Galois Theory by myself and got lost in the following exercise: Let $f$ be an irreducible polynomial of degree $n$, and suppose that the splitting field of $f$ is generated by a ...
1
vote
1answer
22 views

Is this condition enough for irreducibility?

Suppose that $f$ is a polynomial in $Z[X]$, such that $f = (X-\alpha_1) ... (X-\alpha_n)$ with $n$ distinct complex, irrational roots. Suppose that $Q[\alpha_i]$ = $Q[\alpha_1, ..., \alpha_n]$ for all ...
1
vote
2answers
45 views

$x^3+ (5m+1)x+ 5n+1$ is irreducible over $\Bbb Z$

How to prove that the polynomial: $x^3+ (5m+1)x+ 5n+1$ is irreducible over the set of integers for any integers $m$ and $n$? I was trying to put $x= y+p$ for some integer $p$ so that I could apply ...
1
vote
1answer
39 views

How to prove whether $x^{2}+y^{2}+1$ is irreducible over $\mathbb{C}$ or not?

Let's consider a 2-variable polynomial $f(x, y)= x^{2}+y^{2}+1$. It can be established that it's irreducible over $\mathbb{R}$. For example, if it's irreducible over $\mathbb{R}$ as a polynomial of ...
0
votes
2answers
44 views

Question about ideals

I believe the following is a true statement, but I am unsure, so I wanted to check with people. If $p$ is an irreducible polynomial in $n$ indeterminates then $(p)$, the ideal generated by it, is ...
0
votes
1answer
28 views

Proving $f(x)=x^n-p$ is minimal of $\alpha=\sqrt[n]{p}$ over the field F (p is prime)

I am having trouble with the concept of minimal polynomial, In a homework question I have concluded the following: $\mathbb{Q} \le \mathbb{F} \le \mathbb{C}$ - field extensions such that ...
4
votes
2answers
43 views

Example of Field Extension $E/F$ with $Char(F)=2$ and $[E:F]=2$, but is not Galois

I understand that for a field extension $E/F$, if $Char(F)\neq 2$ and $[E:F]=2$ then it must be a Galois Extension. I have proved this, but I am having trouble finding a counterexample when the ...
0
votes
1answer
27 views

Prove that the following polynomial is not dividable over Q

Let $a\neq b$ $|a,b \in\mathbb N$ and let $P(x)=x^3+ax^2+bx+1$ Show that P is irreducible over $\mathbb Q$. Any idea?
0
votes
1answer
27 views

General polynomial of degree $n$ is irreducible from Gauss' Lemma

I'm studying "old-fashioned" Galois theory and the following is an elementary and fundamental problem to keep proceed with my studies. I'm really stuck at that question. Can someone help me please? ...
-1
votes
1answer
49 views

Show any straight line is irreducible

Show that any straight line in $\mathbb{F}^{n}$ is irreducible, where F is an infinite field. I know V($ax+b$) would be a variety that represents any straight line and then V is irreducible if I(V) ...
0
votes
1answer
66 views

I have to show this polynomial is irreducible. [closed]

Suppose that $p(x)=x^9+x^8+x^4+x^2+1 \in \mathbb{Z}_2[x]$. I have to show this polynomial is irreducible.
1
vote
0answers
26 views

Proof Verification of Sum of Irrational Numbers

The chosen answer here claims that for rational $u$ and $v$, $u^{\frac{1}n} + v^{\frac{1}n}$ is rational iff $u^{\frac{1}n}$ and $v^{\frac{1}n}$ are both rational. However, the link to a proof seems ...
1
vote
0answers
16 views

Polynomial Irreducibility prime exponent

Prove that if $p>2$ is prime then $X^p-X-p$ is irreducible over the integers. I tried assuming cases, like the constant term in the factors are 1, -p or -1, p, but can't see how to proceed. (also ...
2
votes
3answers
88 views

Why is $y^2-x^3\in \mathbb{C}[x,y]$ irreducible?

How can I prove that $y^2-x^3\in \mathbb{C}[x,y]$ is irreducible?
0
votes
0answers
51 views

Showing that $x^n+x+3$ is irreducible [duplicate]

How do you show that $x^n+x+3$ is irreducible for all $n \ge 2$?
4
votes
1answer
32 views

Counting total number of monic irreducible polynomials of all degrees $k$ that divide $m$.

Why is the following relation counting monic irreducible polynomials of all degrees $d$ that divide $m$ true? \begin{equation} \sum_{d\ |\ m}\left(\frac{1}{d} \sum_{c\ |\ d} \mu(d/c)\ p^{c}\right) = ...
4
votes
2answers
65 views

Explicitly construct a field with 729 elements

I would like to construct a field with 729 elements. I know that $729 = 3^6$, and that I have to find an irreducible monic polynomial of degree 6 over $GF(3)$. I chose the polynomial $x^6 + 2x^2 + 1$, ...
1
vote
4answers
75 views

Show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$

How do I show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$? Someone says I should use the rational root test, but I don't exactly know how that applies. Thanks for any input.
1
vote
1answer
32 views

Irreducibility of cyclotomic polynomials of prime order

I am stuck with an exercise, where I have to prove the $\Phi_5[X] \in \mathbb{F}_2(x)$ is irreducible. I know that $X^5-1=\Phi_5(X)(X-1)$ (shown in previous part of the exercise) $X^2+X+1$ is the ...
0
votes
1answer
42 views

Galois group of quintic polynomial with 4 complex solution

Suppose we have an irreducible quintic polynomial $f(x)\in \mathbb{Q}[x]$ with 4 complex solutions, say for e.g. $x^5+x^4+x^3-2x^2-2x+5$. It is easy to see that the Galois group $Gal(E/\mathbb{Q})$ ...
0
votes
1answer
21 views

Polynomial divisibility of a minimum polynomial

I am considering $x^2-x-1$ over $F=\mathbb{Z}/2=\{0,1\}$ as the minimum polynomial of a $3\times 3$ matrix $A$ with entries from $F$. I know that the characteristic polynomial (in this case a degree ...
0
votes
0answers
83 views

Factorising $x^7-7x^6+21x^5-35x^4+35x^3-21x^2+20x+14$ into $\Bbb Q$-irreducible factors (Eisenstein's Criterion)

Factorize $x^7-7x^6+21x^5-35x^4+35x^3-21x^2+20x+14$ into $\Bbb Q$-irreducible factors. I've made the substitution $y=x-1$. So I get $y^7+13y+28$ which satisfies Eisenstein's Criterion for $p=13$. ...
1
vote
1answer
54 views

Show for an irreducible polynomial $f(x) \in F[x]$ of degree $n$, $n$ divides $[E:F]$ where $E/F$ is the splitting field of $f(x)$

I want to show that for an irreducible polynomial $f(x)$ in $F[x]$ of degree $n$, and for a splitting field $E/F$ of $f(x)$, $n$ $|$ $[E:F]$. Can anyone provide any hints for me to think about this? ...
6
votes
1answer
141 views

Show that $x^n + x + 3$ is irreducible for all $n \geq 2.$

So first of all, I used the following from my lecture notes: If $f \in \mathbb{Z}[x]$ is primitive (gcd of all the coefficients is 1) - $f$ irreducible in $\mathbb{Z}[x] \Leftrightarrow$ f ...
0
votes
1answer
21 views

Irreducibility of polynomials of a certain kind

Let us look at factorization over the integers of polynomials of the form $x^n+n$. For the first few values of $n$ we get $x+1$ - irreducible $x^2+2$ - irreducible $x^3+3$ - irreducible $x^4+4$ - ...
0
votes
0answers
30 views

If $\frac{m}{n}$ is a root of $a_{a}+a_{1}x+\cdots+ a_{r}x^r$, $m,n$ relatively prime,deduce that if $a_{r} = 1$, then $\frac{m}{n} is an integer$ [duplicate]

I know that if $a_{r} = 1$,then we shall have a monic polynomial with $\frac{m}{n}$ as the root.I don't know how to proceed from this point.
1
vote
2answers
44 views

Show that a polynomial $f(x)$ over a field $k$ is irreducible if the polynomial $f(x+1)$ is irreducible

I was thinking of using contradiction by assuming that that $f(x)$ is reducible but I really don't know how to continue from that idea.
0
votes
1answer
32 views

Polynomial in $Q[x]$ with 2 complex roots

I need to show that for any $n$, I need to show that there is an irreducible polynomial in $Q[x]$ of degree $n$ having exactly $n-2$ real roots. As a hint I have that if $f(x) \in \mathbb{R}[x]$ is a ...
2
votes
1answer
79 views

Irreducible polynomial in $\mathbb Q[x]$ of degree $n$ having exactly $n-2$ real roots.

I need to show that for any $n$ there is an irreducible polynomial in $\mathbb Q[x]$ of degree $n$ having exactly $n-2$ real roots. I know that from a previous exercise that if $f(x) \in ...
2
votes
2answers
38 views

The irreducibility of polynomials for specific cases

I'm trying to prove the following are irreducible over $Q$. $f = 7x^4-18x^3+6x^2-24x+12$ and $g = 2x^3-5x+25$ For the first I have said that f is primitive and used Eisenstein's criterion with ...
0
votes
0answers
22 views

Given $\omega_i$ primitive $n$th roots of unity, prove $\prod_{i=1}^{\phi(n)} (x-\omega_i)$ is the minimal polynomial of $\omega_i$

I'm reading a book about Galois theory which quoted a conclusion without proof: Given primitive $n$th roots of unity $\omega_1, \dots, \omega_{\phi(n)}$, define $$\Phi_n(x) := \prod_{i=1}^{\phi(n)} ...
2
votes
1answer
86 views

Irreducibility of $1+x+\dots+x^{n-1}$ over $\mathbb{F}_2[x]$

Can someone provide a reference of the proof (or the proof itself) of this statement? The polynomial $1+x+\dots+x^{n-1}$ is irreducible over $\mathbf{F}_2[x]$ if and only if $n$ is an odd prime and ...
0
votes
0answers
33 views

Show that there exist an irreducible monic polynomial in $GF(p)[X]$ of degree n.

Let $p$ be a prime integer and let $n$ be a positive integer. Show that there exist an irreducible monic polynomial in $GF(p)[X]$ of degree n. I have seen to many topic abouth how many exists and ...
0
votes
1answer
67 views

Why doesn't there exist a ring homomorphism between these two rings?

Why doesn't there exist a ring homomorphism between $\mathbf{Q}[x]/(x^2-2) $ and $\mathbf{Q}[x]/(x^2 -3) $? I see both rings are in fact fields as the polynomials are irreducible, further I know for ...
1
vote
4answers
41 views

Complex roots of irreducible cubic in $\mathbb{Q}[x]$

Let $$f(x) = x^3 +ax^2 + cx + d \in \mathbb{Q}[x] $$ with one real root, and two complex roots: α and β α and β are conjugates. My task is to show that: $$β \notin \mathbb{Q}(α)$$ I'm confused as I ...
1
vote
1answer
57 views

The polynomial $(X^2+2)^n+5(X^{2n-1}+10X^n+5)$ is irreducible in $\mathbb Z[X]$

Prove that for any postive integer $n$, the polynomial $$(X^2+2)^n+5(X^{2n-1}+10X^n+5)$$ is irreducible in $\mathbb Z[X]$. I have try use Eisenstein's criterion and can't it
1
vote
1answer
119 views

How to prove that $x^9-9x^7+27x^5-30x^3+9x-1$ is irreducible in $\mathbb{Q}[x]$?

The problem says: Given the irreducible polynomial $x^3-3x-1$ with root $2\cos(\pi/9)$, prove that $2\cos(\pi/27)$ is a root of a monic irreducible polynomial of degree 9 over $\mathbb{Q}$, and hence ...