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

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61 views

How to factor polynomials in $\mathbb{Z}_n$?

How to factor a certain polynomial over $Zn$. for example factor the following polynomial into irreducible polynomials in $Z5$: $X^3+X^2+X-1$ or factor the following polynomial into irreducible ...
2
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2answers
43 views

Show that polynomial is reducible

Show that $p(x)$ = $x^3 + 3x^2 + 2x + 4$ is reducible in $\mathbb{Z}$$/$$7$$\mathbb{Z}$ Is the approach for this to factor it and then find a root? I'm a little confused on how to start. Any tips ...
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2answers
37 views

Getting the multiplicative inverse of a polynomial

I have a polynomial $m(x)= x^2 + x + 2$ that's irreducible over $F=\mathbb{Z}/3\mathbb{Z}$. I need to calculate the multiplicative inverse of the polynomial $2x+1$ in $F/(m(x))$. I'd normally use ...
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1answer
44 views

Irreducible in $\Bbb Q[x]$

Suppose $f(x)$ is an polynomial of integer coefficients. If for infinitely many integers $x$, $f(x)$ is prime. Show that $f(x)$ is irreducible in $\Bbb Q[x]$. Suppose $f(x)$ is is reducible in $\Bbb ...
50
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3answers
679 views

A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...
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0answers
13 views

Sufficient condition for a polynomial to split

I found a problem that reads: let $K\subseteq L$ be two fields, and consider an irreducible $f(x)$ in $K[x]$. Show that if there exists an $a\in L$ such that $a$ and $a^2$ are roots of $f$, then $f$ ...
3
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2answers
60 views

Show that $\mathbb Q(\sqrt p) \not\simeq\mathbb Q(\sqrt q)$

I'd like to show that for $p,q$ distinct primes, the extensions $\mathbb Q(\sqrt p),\mathbb Q(\sqrt q)$ are not isomorphic. I don't really have knowledge of the "high-level language" of algebraic ...
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0answers
34 views

Irreducibility of $~\frac{x^{6k+2}-x+1}{x^2-x+1}~$ over $\mathbb Q[x]$

The Artin—Schreier polynomial $~x^n-x+1~$ is always irreducible over $\mathbb Q[x]$, unless $n=6k+2$, in which case it seems to have only two factors, one of which is always $x^2-x+1$. The ...
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1answer
37 views

NTRU cryptosystem

In the NTRU cryptosystem we are dealing with convolution polynomial rings and we compute $f(x)= T(d+1,d)$ and $g(x)= T(d,d)$ but when calculating their inverse in $R_q=(Z/qZ[x] / (x^N-1))$ and ...
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2answers
53 views

Prove particular quintic is irreducible

The problem is to prove that the quintic $$x^5+10x^4+15x^3+15x^2-10x+1$$ is irreducible in the rationals. I don't have much knowledge in group theory, and certainly not in Galois theory, and I'm ...
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0answers
45 views

Factorization and Roots of the following Polynomials

I'm struggling with this exercise $(a)$$f:=T^4 +6T^2 -8T - 3 \in \mathbb{Q}[T]$. Show that f is irreducible, with exactly $2$ real roots. $(b)$ Let $\alpha$ und $\beta$ be two real roots of ...
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votes
1answer
22 views

Show that this polynomial is irreducible over Q

I have to show that the polynomial $1+x^p+x^{2p}+...x^{p^2-p}$ where p is a prime is irreducible over rationals. I am only looking for a hint. How should I go about this?
3
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2answers
77 views

Why is $T^4+6T^2-8T-3\in\mathbb Q[T]$ is irreducible?

Why is $T^4+6T^2-8T-3\in\mathbb Q[T]$ is irreducible ? Obviously Eisenstein doesn't work, but can we do a substitution and obtain a form such that the criterion is applicable ? We didn't cover ...
1
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1answer
46 views

Is $\mathbb{Q}(\pi)$ a simple extension of $\mathbb{Q}\left(\frac{\pi^3}{1+\pi}\right)$?

In the case of an algebraic extension, I could think easier than this case. But I got stuck in this problem. I know that the dimension of $\mathbb{Q}(\pi)$ over ...
1
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1answer
21 views

Irreducible in $F[x]$ implies irreducible in $R[x]$?

Let $R$ be a unique factorization domain, let $F$ be the field of fractions of $R$ and let $f(x)\in R[x]$. I want to show that if $f(x)$ is irreducible in $F[x]$ then $f(x)$ is irreducible in $R[x]$. ...
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votes
1answer
27 views

$p(x)$ divides the minimal polynomial iff $\exists v\ne 0: p(T)(v)=0$

Let $V$, a finite dimensional space. Let $T:V\to V$ a linear transformation. Show that $p(x)$, an irreducible polynomial divides $m_T$ (The minimal polynomial of $T$) iff there is a $V\ni v \ne 0$ ...
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votes
2answers
68 views

Solvability of Artin-Schreier Polynomial

I'm having a hard time trying to prove that the polynomial f(x) = x^p - x - 1 in Z_p[x] is not solvable by radicals even though its Galois Group is solvable. So far, I have shown that the ...
3
votes
1answer
42 views

Galois groups of quintics

I'm trying to determine which subgroups of $S_5$ occur as the Galois group of an irreducible quintic $f\in\Bbb{Z}[X]$. I know such a subgroup of $S_5$ should be transitive, leaving only five ...
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1answer
128 views

Polynomial roots and finite fields

QUESTION: Suppose you are given a polynomial with integer coefficients. And assume it only has complex roots(no real roots). Does it necessarily follow that in a finite field(say mod $p$) that it ...
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3answers
57 views

Is $X^8+1$ reducible in $\mathbb R[x]$?

How to check irreducibilty of $X^{2n}+1$ in $\mathbb R[x]$? $n\geq 2$ I dont know how to do this .Any help
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1answer
41 views

having prime degree:being irreducible or having a root! [duplicate]

Let $p$ be a prime number. Prove that for any field $K$ and any $a \in K$, the polynomial $x^p−a$ is either irreducible, or has a root. it doesn't seem hard,but i have no idea. any hint is welcomed! ...
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2answers
58 views

How do I find out if a polynomial is irreducible?

I have this polynomial: $f(x)=x^4+x^3-4x^2-5x-5$. How can I find out if this polynomial is irreducible over the field $Q$ of rational numbers? I know about mod p irreducibility test but it fails in ...
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3answers
111 views

Irreducible factors of x^16 - 1 over GF(3)

Just want to double check my work. I'm trying to list the irreducible factors of $x^{16} − 1 $ over $GF (3)$ of degree $1$ and $2$ . Here's what I have: $$x + 1, x + 2, x^2 + x + 2, x^2 + 2x + 2$$ ...
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0answers
34 views

Is $f=t^5 + t^4 + 1$ reducible or irreducible over the field of $Z_2$ integers modulo 2.

I need to find out whether $f=t^5 + t^4 + 1$ is reducible or irreducible over the field of $Z_2$ integers modulo 2. I approached the question by substituting 0 and 1 into the function and got answers ...
2
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1answer
68 views

Which one is a field?

Which one is a field? i) $\cfrac{\mathbb{Z}[x]}{\langle{x^2+2}\rangle}$ ii)$\cfrac{\mathbb{Q}[x]}{\langle{x^2-2}\rangle}$ I think both are correct because for both the cases ...
1
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1answer
66 views

Irreducibility over $\mathbb{C}$ of symmetric polynomials

Problem. Find all elementary symmetric polynomials that are irreducible over $\mathbb{C}$. My attempt. It's easy to see that if we have polynomial $f(x_1, \dots, x_n)$ and it can be reduced to ...
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4answers
83 views

Is $x^8+1$ irreducible?

How to decide that $f(x)=x^8+1$ is irreducible or not in the following fields: 1) $F=\mathbb R$ 2) $F=\mathbb Q$ I can't use Eisenstein's criterion. So the only possibility is computing the complex ...
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0answers
29 views

Degree of extension

Let $a, b \in \mathbb{C}$ and let $[\mathbb{Q}(a) : \mathbb{Q}]=m$, $[\mathbb{Q}(b) : \mathbb{Q}]=n$. Show that $[\mathbb{Q}(a,b):\mathbb{Q}]\leq mn$. If $(m,n)=1$ show that ...
2
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1answer
36 views

Irreducible polynomial/Splitting field

Let $f(x)=x^4+16 \in \mathbb{Q}[x]$. Split $f(x)$ into a product of first degree polynomials in $\mathbb{C}[x]$. Show that $f(x)$ is an irreducible polynomial of $\mathbb{Q}[x]$. Find the splitting ...
0
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1answer
37 views

Show that $p=2^k+1$

When $p$ is an odd prime and $a=Re \left ( e^{\frac{2 \pi i}{p}} \right)$ then $[\mathbb{Q}(a) : \mathbb{Q}]=\frac{p-1}{2}$. Let $\theta = \frac{2 \pi}{p}$. If $\sin{\theta}$ is a constructable ...
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1answer
15 views

What is the relation between $Irr(a, F)$ and $Irr(a, K)$?

We have that $F \leq K \leq L$ and $a \in L$. If $a$ is algebraic over $F$ then it is also algebraic over $K$. What is the relation between $Irr(a, F)$ and $Irr(a, K)$? Let $Irr(a, K)=p(x) \in K[x]$ ...
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2answers
68 views

$\sqrt{5} \in \mathbb{R}$ is algebraic over $\mathbb{Q}$

I have to show that $\sqrt{5} \in \mathbb{R}$ is algebraic over $\mathbb{Q}$ and then I have to find $Irr(\sqrt{5}, \mathbb{Q})$. How can I show that $\sqrt{5} \in \mathbb{R}$ is algebraic over ...
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1answer
20 views

How do we find the embeddings?

In my notes there is the following example: $$\mathbb{Q}(\sqrt{2}) \overset{\widetilde{\sigma}}{\longrightarrow}\mathbb{R}\\ | \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | \\ \mathbb{Q} \overset{\sigma=id ...
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1answer
43 views

The union of finite field extensions is a finite field extension

Assume that all elements under discussion are algebraic over $F$. Let the notation "$K=F(A)$" mean that $A\subseteq K$ and there is an injective homomorphism $\sigma:F\to K$, and every element of $K$ ...
2
votes
4answers
126 views

Roots of different irreducible polynomials are algebraically independent

Let $F$ be a field, and let $f$ be a monic irreducible polynomial over $F$. Let $\alpha$ be a root of some other monic irreducible $g\ne f$. Then is $f$ still irreducible in $F(\alpha)$? Is it true ...
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2answers
30 views

How to show that an ideal of $F[x]$ containing an irreducible polynomial of degree $n$ and a nonzero polynomial of degree $<n$ is $F[x]$?

Let $F$ be a field and suppose that $I$ is an ideal of $F[x]$ which contains an irreducible polynomial of degree $n$ and a nonzero polynomial of degree less than $n$. Show that $I=F[x]$. I can't ...
3
votes
4answers
82 views

Prove irreducibility of $X^4+15X^3+7\in\mathbb Z[X]$

What technique can I apply to prove that $X^4+15X^3+7\in\mathbb Z[X]$ is irriducible? I can't apply Eisenstein because 7 and 15 have no common prime multiple but 1; I tried to apply Eisenstein with ...
1
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2answers
54 views

Factorize x^3+3

Task: Factorize $x^3+3$ in $\mathbb{R}$ and $GF(7)$ I think the solution is $x^3+3$ in both cases, so the polynomial already is irreductible. Is my assuption is right, how do i show that?
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Irreducibility over Q doesn't imply irreducibility over R

I want a counterexample of polynomial that is irreducible over $\mathbb Q$ but not irreducible over $\mathbb R$ (i.e not maximal over $\mathbb R$).
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GCD of polynomials in $\mathbb{F}_2[x]$

How can one show that $\gcd(x^4+x+1,x^4-x)=1$ in $\mathbb{F}_2[x]$? Is it because the polynomials do not share any irreducible monic polynomials i.e. $1, x, x+1$ as factors?
2
votes
1answer
15 views

Irreducibility over a field of rational functions

Let $K$ be a field, and let $f,g\in K[Y]$ be coprime with $\deg(fg)\geq1$. How do I prove that $f-gX\in K(X)[Y]$ is irreducible? I tried the "generic" approach of assuming the existence of a ...
0
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2answers
15 views

Show $X_1^5+X_1^2X_2+X_1X_2+X_2$ is irreducible

For a field $K$, is the polynomial $X_1^5+X_1^2X_2+X_1X_2+X_2$ irreducible in $K[X_1,X_2]$? I think I have to show it for $K(X_1)[X_2]$ but I don't know how to do that.
0
votes
1answer
44 views

$X^3+2$ is irreducible in $\mathbb{F}_7[X]$

Let $I=(X^3+2)$ be the principal ideal of $\mathbb{F}_7[X]$ generated by $X^3+2$. Show that $X^3+2$ is irreducible in $\mathbb{F}_7[X]$. Can someone give me the first step on how to do this ...
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3answers
32 views

Prime ideals in $\mathbb{Q}[X]$

Could you tell me why prime ideals in $\mathbb{Q}[X]$ are of the form $(q(x))$ where $q \in \mathbb{Q}[X]$ is irreducible?
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2answers
64 views

Determine whether the polynomial $x^2-12$ in $\mathbb Z[x]$ satisfies an Eisenstein criterion for irreducibility over $\mathbb Q$

Determine whether the polynomial $x^2-12$ in $\mathbb Z[x]$ satisfies an Eisenstein criterion for irreducibility over $\mathbb Q$. So If I understand correctly, I start with $x^2=12$ and then you ...
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2answers
57 views

Question on reducibility over rationals.

If we have a polynomial with real coefficients $P(x)$ for which $P(x)=q$ has a rational solution for all rational numbers q, does that mean that the coefficients actually have to be rational? I think ...
2
votes
0answers
40 views

$x^5+a$ is reducible in $\mathbb{Z}_5[x]$ for each $a\in\mathbb{Z}_5$

Question: Show that $x^3+a$ is reducible in $\mathbb{Z}_3[x]$ for each $a\in\mathbb{Z}_3$, and that $x^5+a$ is reducible in $\mathbb{Z}_5[x]$ for each $a\in\mathbb{Z}_5$ So I got these two as my ...
0
votes
2answers
24 views

Field extensions and minimal polynomial

I have to find the degree of $1+\sqrt[3]{2}+\sqrt[3]{4}$ over $\mathbb{Q}$. This is what I found already: $\mathbb{Q}(1+\sqrt[3]{2}+\sqrt[3]{4})=\mathbb{Q}(\sqrt[3]{2},\sqrt[3]{4})$ ...
2
votes
0answers
35 views

The number of bijective polynomials of particular degree in a field

I need to know please: In a finite field of q elements how many bijective polynomials exist whose degree are smaller than d?
1
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2answers
59 views

Irreducible polynomials in two and four variables

I am trying to show that the polynomials : 1) $X^2+Y^2-1$ 2) $XT-YZ$ are irreducible in $\mathbb Q[X,Y]$ and $\mathbb Q[X,Y,Z,T]$ respectively. For 1) I know that $\mathbb Q[X,Y]=(\mathbb ...