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Questions tagged [irreducible-polynomials]

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

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Is the area enclosed by p(x,y) always irrational?

Take a polynomial $p \in \mathbb{Q}[X,Y]$. Now draw the graph of $p(x,y)=0$. If, like $X^2-Y^2-1$, this turns out to enclose a finite area, is the area enclosed always irrational? There are some ...
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Prove that $x^6+5x^2+8$ is reducible over Z (integer)?

$attempts:-$ 1] I tried to replace $X^2=t$ but nothing click after that . 2] then I tried to replace this polynomial say P(x) by P(x+1) or P(x-1) to apply Eisenstein's Irreducibility Criterion Theorem ...
JAYENDRA JHA's user avatar
1 vote
2 answers
105 views

Proposition 8 Corollary 1, Section 5.7 of Hungerford’s Algebra

Corollary 1.9. Let $E$ and $F$ each be extension fields of $K$ and let $u\in E$ and $v\in F$ be algebraic over $K$. Then $u$ and $v$ are roots of the same irreducible polynomial $f \in K[x]$ if and ...
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Determining the Equality of Two Field Extensions

Let $F$ be a field of characteristic $0$. Let $F(\alpha)/F$ be a finite extension of degree not divisible by $3$. Is is true that $F(\alpha^3)=F(\alpha)$? If we assume that they are not equal, since $\...
Ty Perkins's user avatar
1 vote
1 answer
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Field of positive characteristic not perfect in which every polynomial is reducible [closed]

It is well-known that, in a real closed field $K$, every polynomial of degree>2 is reducible in $K$. But in this case the characteristic of $K$ is zero. My question is: there exists a field $F$ ...
Medo's user avatar
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Why p mod irreducibility test not working here?

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 reducing all ...
math student's user avatar
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2 votes
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If $F/K$ is normal extension and $f \in K[x]$ irreducible and $f=\prod_{i=1}^{n}g_{i}^{m_{i}}$ in $F[x]$ then $m_{i}=m_{j}$ for all $i,j$

So I have that question: Let $F/K$ be a normal extension and $f$ irreducible polynomial in $K[x]$ assume that $f=\prod_{i=1}^{n}g_{i}^{m_{i}}$ where $g_i$ is irreducible in $F[x]$ ($m_i \geq 1$) then ...
oneneedsanswers's user avatar
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irreducible polynomial of $\alpha$ in $K$ over $F$ is unique, but why the trace of $\alpha$ in $L$ $\supset$ $K$ is sum of more automorphism images?

Can you tell me this is false? Or, is the field Trace map only defined on specific type of field extension? And, what is the type of extension? I'm curious about Definition of serge lang. $Tr^E_K$ ($\...
Snailman's user avatar
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Polynomial reduction modulo n. Irreducible polynomal.

I have the following polynomial: $f(x)=x^4+1$. I have to prove that it is irreducible over $\mathbb{Z}[x]$ using reduction criterion. The Reduction Criterion says that: Let $\mathfrak{m}$ be maximal ...
ITChristian's user avatar
1 vote
0 answers
46 views

Irreducibility of the $p^k$-th cyclotomic polynomial

I want to prove that the cyclotomic polynomial $\Phi_{p^k}$ is irreducible using Eisenstein (I know that every cyclotomic polynomial is irreducible, I am just trying this approach). I am exposing what ...
lkksn's user avatar
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1 answer
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To determine the number of roots for all antiderivative of a cubic polynomial

Let $f(x)$ be a cubic polynomial with real coefficients. Suppose that $f(x)$ has exactly one real root which is simple. Which of the following statements holds for all antiderivative $F(x)$ of $f(x)$ ?...
user-492177's user avatar
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3 votes
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An efficient algorithm for determining whether a quartic with integer coefficients is irreducible over $\mathbb{Z}$

I'm interested in what efficient algorithm could be used for determining if a quartic polynomial with integer coefficients is irreducible over $\mathbb{Z}$. For quadratics and cubics it's not too bad, ...
Robin's user avatar
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1 vote
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Let $I=(x^2+x+1)$. Is $\Bbb Z_3[x]/I$ an integral domain?

Let $I=(x^2+x+1)$. Is $\Bbb Z_3[x]/I$ an integral domain? My solution goes like this: If $\Bbb Z_3[x]/I$ is an integral domain then $I$ is a prime ideal. But $I$ is a non-zero prime ideal and since, ($...
Thomas Finley's user avatar
2 votes
1 answer
125 views

Irreducibility of a Polynomial with Prime Exponents

Let $f(x) = (x^p - a_1)(x^p - a_2) \ldots (x^p - a_{2n}) - 1$ where $a_i \geq 1$ are distinct positive integers where at least two of them are even, and $n \geq 1$ is a positive integer and $p$ is ...
math.enthusiast9's user avatar
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2 answers
119 views

How to prove: $f\equiv0\,(\mathrm{mod}\,p^2)\iff f'\equiv0\,(\mathrm{mod}\,p)$? [closed]

Edit: Corrected the mod order. It might be trivial, but I have no idea at all about it. For a univariate polynomial $p$, then $f\equiv0\,(\mathrm{mod}\,p^2)\iff f'\equiv0\,(\mathrm{mod}\,p)$ where $f'$...
MathArt's user avatar
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1 vote
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Factorize the polynomial $p(x)=x^4+x^3+(1+i)x^2+(1-i)x+3i$. [closed]

I stumbled upon the question that 'Factorize the polynomial $$p(x)=x^4+x^3+(1+i)x^2+(1-i)x+3i$$ It is commonly known that $\mathbb C$ is algebraically closed. So, any polynomial has at least one ...
Fuat Ray's user avatar
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Reducibility of $x^2-7$ over $\mathbb{Q}(\sqrt[5]{3})$

Suppose for a contradiction that $x^2-7$ is reducible over $\mathbb{Q}(\sqrt[5]{3})$. Then $\sqrt{7}\in\mathbb{Q}(\sqrt[5]{3})$. It follows that $\mathbb{Q}\subset\mathbb{Q}(\sqrt{7})\subset\mathbb{Q}(...
spinosarus123's user avatar
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Showing that $x^4+2x^2+5$ is irreducible over rational numbers [duplicate]

I want to show that $P(x)= x^4+2x^2+5$ is irreducible over rational numbers. I have decomposed the polynomial into $(x^2+ax+b)(x^2+cx+d)$, and since $P(x)$ is an even function, we have either $P(x)=(x^...
Etemon's user avatar
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1 vote
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Number of irreducible polynomials of degree at most n over a finite field

We know that the number $N(n,q)$ of irreducible polynomials of degree $n$ over the finite field $\mathbb{F}_q$ is given by Gauss’s formula $$N(n,q)=\frac{q-1}{n} \sum_{d\mid n}\mu(n/d)q^d.$$ The number ...
Hassen Chakroun's user avatar
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1 answer
52 views

Understanding role of units in definition of irreducible element

According to link, The polynomial $f(x) = 2x^2 + 4=2(x^2+2)$ is irreducible over $\mathbb{Q}$ but reducible over $\mathbb{Z}$, neither $2$ nor $x^2 + 2$ is a unit in $\mathbb{Z}[x]$. First, I am ...
Fuat Ray's user avatar
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1 vote
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If polynomial is irreducible over finite field $\mathbb F_p$ then it is irreducible over $\mathbb Z$.

According to link, It is easy to show its contrapositive which is If the polynomial is reducible over $\mathbb Z$ then it is reducible over finite field $\mathbb F_p$ of order $p$. Let $f(x)=g(x)h(x) \...
Fuat Ray's user avatar
  • 1,140
2 votes
1 answer
70 views

If $f(x)\in \mathbb{Z}[x]$ is irreducible (over $\mathbb{Q}$), is it always possible to find $a$ and $b$ in $\mathbb{Q}$ with $f(ax+b)$ Eisenstein? [duplicate]

My initial thought is no, simply because it seems too easy if it is true. The simplest example of a nontrivial irreducible polynomial I could think of was $f(x)=x^2+1$. Unfortunately, $f(x+1)$ is ...
ljfirth's user avatar
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2 votes
1 answer
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Finding the irreducible elements of a polynomial ring

Let $R = \{f(X)\in \mathbb{Q}[X] \space | \space f(0)\in \mathbb{Z}\}$. I am asked to find the irreducible elements of $R$. I have found that the units in $R$ are $\pm 1$. If $\deg f = 0$, i.e. $f$ is ...
idk31909310's user avatar
2 votes
1 answer
53 views

$X^{6}+(T-22) X^{4}+T X^{3}+(T-22) X^{2}+1$ is irreducible over $\mathbb Q(T)$

I'm asked to prove that $f(X)=X^{6}+(T-22) X^{4}+T X^{3}+(T-22) X^{2}+1$ is irreducible over $\mathbb Q(T)$. For this, I need first to prove that the polynomials $p(X) = X^{6}-22 X^{4}-22 X^{2}+1$ and ...
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Exercise 4, Section 5.3 of Hungerford’s Algebra

Hungerford, Algebra, page 257, gives the following as Definition 3.1: Let $S$ be a set of polynomials of positive degree in $K[x]$. an extension field $F$ of $K$ is said to be a splitting field over $...
user264745's user avatar
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Artin's theorem exercise - proving that the fixed field is generated by the coefficients of the minimal polynomial

Suppose $L/K$ is a finite extension. $G$ is a finite group of $K$-automorphisms of $L$. Denote by $L^G$ the field elements of $L$ fixed by action of $G$. For any $\alpha \in L$ we write $f(t, \alpha) =...
Featherball's user avatar
3 votes
1 answer
72 views

For what integers $m\gt n\gt 0$, the polynomial $x^m+x^n+1$ is irreducible over $\mathbb Q$?

I came up with this problem and have found it interesting. Problem. For what integers $m\gt n\gt 0$, the polynomial $f(x)=x^m+x^n+1$ is irreducible in $\mathbb Q[x]$? If $mn\equiv 2 \pmod 3$, i.e. one ...
Cyankite's user avatar
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1 vote
1 answer
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Irreducibility over Field extensions.

This is part of something bigger that I'm trying to prove, but I'm having difficulties with this part, which seems like it should be relatively simple. The polynomial $(x^2 - p)$, $p$ prime, is ...
lkksn's user avatar
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Algebraic set of irreducible polynomial is irreducible?

I'm going through Fulton's Algebraic Curves, and just completed an exercise showing that $V(Y-X^2)$ is irreducible. My solution for this was to show that $Y-X^2$ is irreducible in $k[X,Y]$ (by showing ...
Trettman's user avatar
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2 answers
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Is this $f(x) = x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$ irreducible in GF(5)?

Perhaps one can somehow apply Eisenstein's sign here by considering $f(x+1)$, but by default it is formulated for the expansion over $\mathbb{Q}$ of a polynomial from $\mathbb{Z}[x]$. Here we have $GF(...
mackenzie's user avatar
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1 answer
30 views

Prove: either there exists a $ j $ such that $ |x_j| > 1 $, or $ |x_j| = 1 $ for all $ j $.

Let $ f(x) = (x - x_1) \cdots (x - x_n) $ be an $ n $-degree monic irreducible polynomial with integer coefficients. Prove: either there exists a $ j $ such that $ |x_j| > 1 $, or $ |x_j| = 1 $ for ...
lux fun's user avatar
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1 vote
0 answers
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Find a monic irreducible polynomial equivelent to $(x-x_1)(x-x_2)\Phi_m$

Find a monic irreducible polynomial $f(x) = (x - x_1) ... (x - x_n)$, $|x_1| > 1$ and $x_1$ is real, |x_2| < 1 and $x_2$ is real, $|x_j| = 1$ for all $j > 2$. And First, prove $n > 3$ ...
lux fun's user avatar
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0 votes
0 answers
39 views

Show that $f(t) = t^2 + 4t + 2$ is irreducible over $\mathbb{F_5}$

I had a very short introduction to some exercises about finite fields, but I don't understand the theory very well, so I'm a bit confused. I have this polynomial $t^2 + 4t + 2$ with coefficients that ...
user33's user avatar
  • 141
0 votes
2 answers
65 views

For $k\subset F \subset E$ algebraic field extensions, if "all" irreducible polynomials with a root in E factor in $F[x]$ then $F=E$?

Let $k \subseteq F\subseteq E$ be field extensions with $E$ algebraic extension over $k$. Suppose $\forall a \in E\backslash F$ the irreducible polynomial $p(x)$ of $a$ over $k$, factors non-trivially ...
Cezar's user avatar
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0 answers
24 views

Degrees of irreducible polynomials over finite fields [duplicate]

Artin chapter 15 states the following as a corollary: Corollary 15.7.4 For every positive integer $r$, there exists an irreducible polynomial of degree $r$ over the prime field $\mathbb{F}_p$. ...
Ben Carpenter's user avatar
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0 answers
54 views

When is $f = X^4 -1 \in \mathbb{F}_p[X], p $prime, irreducible and/or seperable? [duplicate]

I'm having some trouble figuring out a solution to this. I understand that $f$ is separable, iff all its roots are distinct, however I'm completely clueless about how to investigate that criterion......
Raiden's user avatar
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1 vote
0 answers
28 views

Monoid structure on polynomial quotient ring / quotient fields

Let $\mathbb F$ be a field and let $P\in\mathbb F[X]$ be an irreducible polynomial. Write $\mathbb F'=\mathbb F[X]/P$, and note that this is also a field. Let $P'\in\mathbb F[X]$ be any polynomial ...
Jim's user avatar
  • 538
4 votes
1 answer
81 views

$x_1,...,x_4$ are irreducible in $k[x_1,..,x_4]/\langle x_1x_4-x_2x_3\rangle$

Let $k$ be a field (not sure whether there are conditions on characteristic and algebraic closure). I want to show that the classes of $x_1,...,x_4$ in $R:=k[x_1,..,x_4]/\langle x_1x_4-x_2x_3\rangle$ ...
Flynn Fehre's user avatar
0 votes
0 answers
61 views

Solving sextic with Kampé de Fériet functions

I recently faced a problem with a polynomial of 6th degree, a sextic. I want an analytical solution to the problem, and I read in the last few days that Kampé de Fériet functions can solve general ...
Eric D'Antona's user avatar
0 votes
1 answer
57 views

Proving the irreducibility of a polynomial in a field extension.

Let $L$ be splitting field of the polynomial $f(x)=(x^3+2x+1)(x^3+x^2+2)(x^2+1) \in \mathbb{F}_3[x]$. How many proper subfield does $L$ have? This is a question from an old qual at my university, and ...
Ty Perkins's user avatar
0 votes
1 answer
32 views

$f$ is irreducible if the polynomial reduced $p$ is irreducible and the degrees are the same

Let $f$ be an irreducible polynomial and $h(f)$ the polynomial with coefficients reduced modulo a prime $p$. Then if $\deg(f)=\deg(h(f))$ and $h(f)$ is irreducible then $f$ is irreducible as an ...
Xaver Wallenstein's user avatar
0 votes
1 answer
45 views

Irreducible polynomials with complex root.

I need to show that if $f$ and $g$ are irreducible in $\mathbb{Q}$[$x$] and they share a common complex root, then there is $a \in \mathbb{Q}$ such that $f = a . g$. What I thought: Call $u \in \...
lkksn's user avatar
  • 121
2 votes
1 answer
104 views

Prove that $x^4+3x^3+3x^2-5$ is irreducible over $\mathbb{Q}$ [duplicate]

I was trying to prove that $x^4+3x^3+3x^2-5$ is irreducible over $\mathbb{Q}$ but I have troubles understanding the solution. The solution reads: Consider the polynomial mod 2: $x^4+3x^3+3x^2-5 \equiv ...
Ishigami's user avatar
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0 votes
0 answers
33 views

Calculating $[K(t):K\left(\dfrac{t^2}{t-1}\right)]$ with $t$ a trascendental element over $K$.

I was asked in an excercise of my fields and Galois theory course to calculate the degree of the extension $K(t)/K\left(\dfrac{t^2}{t-1}\right)$ with $t$ a trascendental over $K$. I started stuying ...
IAG's user avatar
  • 223
7 votes
1 answer
191 views

Factorization and irreducibilty for $x^n-2x^m+1$ trinomials.

I have encountered a weird phenomenon while trying to solve a problem on Reddit. Here is the phenomenon. Let $a>b \in \mathbb{N}$ and $p_{(a,b)} = x^a - 2x^b + 1$ It seems that if $gcd(a,b,c,d) = 1,...
Vatsa Srinivas's user avatar
2 votes
4 answers
70 views

Irreducibility of $X^4-\sqrt{2}$ over $\mathbb{Q}(\sqrt{2})$.

To prove that $X^4-23$ is irreducible in $\mathbb{Q}[X]$ we can do the following: We use the Eisenstein criterion with $a=23$ to see that it is irreducible in $\mathbb{Z}[X]$, and then we conclude ...
IAG's user avatar
  • 223
1 vote
0 answers
25 views

$f$ and $Df$ are not relative primes in $F[X]$, then they are not relative primes in $K[X]$.

The question I will ask originates in the context of the theory of perfect fields and separable extensions, but it is a question of irreducibility of polynomials between extensions of fields and of ...
IAG's user avatar
  • 223
4 votes
1 answer
54 views

Using Eisenstein's Criterion with a transformation

I'm trying to prove that the following polynomial is irreducible in $\mathbb{Q}$ :$$14x^{10} + 18x^9 + 4x^3 + 1$$ Obviously, we can't apply Eisenstein's Criterion here so I tried setting $y = \frac{1}{...
user avatar
4 votes
0 answers
36 views

Function field over a perfect field can be generated by two elements

I have two questions about the following theorem: Theorem: Let $K$ be a perfect field, $F$ a function field in one variable over $K$ (i.e., a finite algebraic extension of $K(t)$). Then there is $x \...
Marktmeister's user avatar
  • 1,648
1 vote
1 answer
48 views

is $x^4-x+1$ irreducible in $\mathbb{Z}_3$

i was wondering if i checked correctly. i found all polynomials in $\mathbb{Z}_3[x]$ of degree 2 which are irreducible and checked if they are divisible without remainder the polynomials i tried were $...
macman's user avatar
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