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Questions tagged [irreducible-polynomials]

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

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Degree of splitting field for $f(x) = x^4 - x^2 + 4$ over $\mathbb{Q}$

I started for finding the roots of the polynomial (4 in total) which took the forms $$ \pm \sqrt{\frac{1}{2}\left( 1 \pm i \sqrt{15} \ \right)} $$ I figured that adjoining the positive square root ...
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0answers
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Question about Lang's Chapter 6 Theorem 9.1

I am an undergraduate working through Chris Hall's result about infinitely many twin irreducible polynomials over finite fields. He begins his argument with a lemma, If $q \equiv 1$ mod $l$ for ...
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1answer
23 views

Applying degree 2 or 3 irreducibility tests to higher degree

Given the polynomial$\ x^4+x+1$, I have to find out if it is irreducible over $\mathbb Q $. When looking at the solutions, they applied the degree 2 or 3 irreducibly tests to determine that it ...
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1answer
64 views

Is $X^8+a \in \mathbb{F}_{49}[x]$ irreducible?

Let $f(x) = x^8+a \in \mathbb{F}_{49}[x]$ with $a \in \mathbb{F}_{49}\setminus \{0\}$. Find all $a$ such that $f$ is reducible over $\mathbb{F}_{49}$! What I know is that $\mathbb{F}_{49} \cong \...
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75 views

A peculiarity of Eisenstein polynomials

I've tested polynomials of degree 6 with random integer coefficients $|a_i|<50$ in test-series of $10,000$. The probability of a random primitive polynomial of the kind to be reducible seems to be ...
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1answer
23 views

The multiplicity of a root $r$ of a irreducible polynomial is a power of $p$ characteristic

$f$ is an irreducible polynomial over a field $K$ of characteristic $p$. $F$ is a splitting field of $f$ over $K$ and $u_1$ a root of $f$. I have shown that $f=[(x-u_1)\cdots (x-u_n)]^{[K(u_1):K]_s}$ ...
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Decide if f(X) is irreducible in the following rings

Let $$f(X) = 78X^3 + 174X^2 − 116 ∈ Z[X]$$ My question is to decide if $f(X)$ is irreducible in $Z[X], Q[X] and R[X]$ I have tried finding a prime number 29, and to fulfil the Eisenstein's ...
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2answers
64 views

Irreducible polynomial in $\mathbb C[x_1,x_2]$ also irreducible in $\mathbb C[x_1,x_2,…x_r]$? [duplicate]

Let $f_1(x_1), f_2(x_2)$ be polynomials in a single variable, of relatively prime degree, with complex coefficients. If $f_1(x_1)+f_2(x_2)$ is irreducible in $\mathbb C[x_1,x_2]$, then is it ...
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1answer
44 views

Irreducibility of the following Polynomial over $\mathbb{Q}$

Take the polynomial $x^4 + 10x^2 + 1$. Is this irreducible over $\mathbb{Q}$? If so, what is the best way to show this? I know it can be rewritten: $$ (x^2 + 5)^2 -24 $$ Which can be simplified over ...
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$x^3 - 3x - 1$ irreducible in $\mathbb Z[x]$ by Gauss Lemma

In Dummit & Foote, they claim this can be shown to be irreducible by Gauss Lemma and applying it to show it has no rational root. But this doesn't make sense to me since Gauss Lemma says: ...
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Determine whether a polynomial is irreducible

Consider the polynomial $P=X^5-X-1\in\Bbb{F}_3[X]$. I want to show that $P$ is irreducible. We can easily check it has no roots, so the only way it could not be irreducible is by being a product of ...
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How to show that a given polynomial is irreducible in a cyclotomic field

I'm beginning to study McCarthy's Algebraic Extensions of Fields, and one of the first problems is to give a factorization of $x^4 + 1$ in $K[x]$, where $K=\mathbb{Q}(a)$ and $a$ is a root of $x^4+1$ (...
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Proof of “Lagrange's Lemma”

Can anyone share a link of proof of the following fact ? Let $f(x) \in K[x] $ be an irreducible polynomial with $n$ distinct roots $r_i$ and let $g(x_1,\dots, x_n)$ and $h(x_1,\dots, x_n)$ be ...
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1answer
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Polynomials taking a prime or 1 value on infinitely many points are irreducible

Let $ P \in \mathbb{Z}[X] $ monic of degree d such that there exists an infinite sequence $ (x_i) \subset \mathbb{Z} $ , where $ |P(x_i)| $ is prime or equal to 1. Show that P is irreducible and that ...
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Irreducible elements in a factor ring

Here is the problem 11.2.16 from Artin, Algebra: $F$ is a field. Let $R$ be the ring $$\frac {F[u,v,y,x_1,x_2,x_3,...]}{(x_1y-uv,x{_2}{^2}-x_1,x{_3}{^2}-x_2,...)}.$$ Show that $u,v$ are ...
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1answer
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Primes $p=n^6+1$

Which is the least odd prime $p=n^6+1$ for some $n\in\mathbb N$? I have tested for $n\leq 10,000$ without finding any. Due to a conjecture of Bunyakovsky there are an infinite number of such primes, ...
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34 views

If $\sigma^{*}\left(p\left(x\right)\right)$ is irreducible in $S\left[x\right]$, then $p\left(x\right)$ is irreducible in $R\left[x\right]$.

Let $\sigma:R\to S$ be a ring homomorphism, and $\sigma^{*}$ be the induced map $\sigma^{*}:R\left[x\right]\to S\left[x\right]$ given by $\sum a_{i}x^{i}\mapsto\sum\sigma\left(a_{i}\right)x^{i}$ (this ...
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Prove that $x^{5}-x+a$ is an irreducible polynomial [duplicate]

Question. Prove that $f(x)=x^{5}-x+a$ is irreducible in $\Bbb Z[x]$ if $5\nmid a$. My approach. If we let $f(x)=(x+a_0)(x^4+b_3x^3+b_2x^2+b_1x+b_0)$, we know that $a=a^5-a$. So $5 \mid a$ and it's ...
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A consequence of Gauss' lemma

The following is written in Fulton's Algebraic curves book but I can't quite understand it. If $R$ is a UFD with quotient field $K$, then (by Gauss) any irreducible element $F\in R[X]$ remains ...
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1answer
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A problem of factoring a polynomial with a hint

PT $(x-a_{1})(x-a_{2})..(x-a_{n})+1$ can not be factored into two smaller polynomial $P(x)$ and $Q(x)$ with integer coefficients, where $a_{i}$'s are all different integers. This problem can be ...
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3answers
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$x^2+y^2-1$ is irreducible in $\mathbb{Q}[x,y]$.

To solve the Dummit-Foote's exercise I've stuck here with this problem : (P-312) Qn11. Show that $p(x)=x^2+y^2-1$ is irreducible in $\mathbb{Q}[x,y]$. This is a polynomial of degree 2. But I can ...
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2answers
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The polynomial $x^6+x^3+1$ is irreducible over $\mathbb{Q}[x]$

To solve Dummit-Foote's exercise I want to check that whether the polynomial $p(x)= x^6+x^3+1\in \mathbb{Z}[x]$ is reducible over $\mathbb{Q}$ or not. I've seen that this polynomial has no zero in $\...
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3answers
83 views

Is the polynomial $x^4+10x^2+1$ reducible over $\mathbb{Z}[x]$? [duplicate]

Is the polynomial $x^4+10x^2+1$ reducible over $\mathbb{Z}[x]$?
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40 views

Irreducibilty of $x^4-x^2-1$ [duplicate]

I was studying in my Field Theory course when I saw in the exercises that the polynomial $p(x)=x^4-x^2-1$ is irreducible over $\mathbb Q$. But there is no proof for it; I tried p-Einstein's method ...
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23 views

Splitting field of an irreductible polynomial $f(X) \in F_{q}[X]$

Let $F_q$ be a finite field ($q$ is a power a prime) and irreductible polynomial $f(X)\in F_q[X]$ with degree $n\geq 2$. I have to see that $F_{q^n}$ is the splitting field of $f$ over $F_q$, and ...
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32 views

irreducibility of polynomial over field of rational functions

Here This is an exercise from Dummit and Foote where the following hint is also given : $\mathbf{(K[X])(Y)=(K[Y])(X)}$. Does this mean that we can consider our polynomial over $\mathbf K[Y]$ with ...
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24 views

How to show that it is irreducible?

Let $\gcd(p,n)=1$. Consider $\;x^n-1$ over $\Bbb F_p[x]\;$. If its splitting field is $K$ find $\;[K:\Bbb F_p]$. Now $K=\Bbb F_p(e^{2\pi i/n})$ Also the polynomial of which $e^{2\pi i/n}$ is a root ...
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1answer
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Factorization of $x^n - x$

While studying about the galois field $GF(2)[x]$, i wanted to find out whether a given polynomial is primitive or not. To do that i need to factor this term: $x^8 - x$. I got the only solution that ...
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1answer
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Does a linear change of coordinate affect the ireducibility of a polynomial?

Let $R$ be an integral domain, and $f\in R[x]$. Is it true that $f(rx+b)$ being irreducible implies that $f$ is irreducible for $r,b \in R$? I know that this is true for $R= \mathbb R$ or $\mathbb C$....
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1answer
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Silly question: are polynomials “X” and “X^2” reducible?

Since f(0)=0 holds for both, are they both reducible polynomials? I'm asking because I'm working on this question where they're asking you to factor the following ...
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Why is $(x^{2n+1} - (2n+1)x^{n+1} + (2n+1)x^n - 1)/(x-1)^3$ irreducible?

Consider the following polynomial $$ f_n(x)=x^{2n+1} - (2n+1)x^{n+1} + (2n+1)x^n - 1 $$ Try the first $n$, I find that the $(x-1)^3$ is its factor: $$ f_1(x)=(x-1)^3,\\ f_2(x)=(x-1)^3 (x^2+3x+1),\\ ...
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1answer
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A conjecture about irreducible polynomials with integer coefficients

Let $f\in\mathbb Z[X]$, define $\operatorname{P}^+(f)$ as the number of primes $>0$ that $f$ assumes at distinct integral arguments. Theorem: If $f\in\mathbb Z[X]$ is non constant and reducible ...
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Is $\langle x_{1}\cdots x_{n}-1 \rangle$ a prime ideal in $\mathbb{C}[x_{1},\dots, x_{n}]$?

I've heard from Google that the algebraic torus, the zero locus of $x_{1}\cdots x_{n}-1=0$, is an affine variety, which means $x_{1}\cdots x_{n}-1$ is an irreducible polynomial, which means $\langle ...
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1answer
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Meromorphic Functions on Riemann Surfaces

My question refers to a step in the proof of Prop. 3.3.5 Szamuely and Tamásin's "Galois groups and fundamental groups": Here the statement and Thm 3.3.3 & lemma 3.3.6: The main ingredients for ...
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1answer
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Invertible elements of $\mathbb{Z}_3[x] / (x^4+x^3-1)^3$

Let $F=\mathbb{Z}/3\mathbb{Z}$, $h(x)=x^4+x^3-1$, $R = F[x]/(h(x)^3)$. I know $R$ has $4$ ideals and $1$ maximal ideal. Let $M$ be the maximal ideal $(h(x))/(h(x)^3)$ I need to find the number of ...
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1answer
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Algebraic Field Extensions and Irreducible Polynomials

Let $E/K$ be a field extension, $a,b\in E$ algebraic over $K$. Show: $\text{Min}(a,K,X)$ irreducible over $K(b)$ if and only if $\text{Min}(b,K,X)$ irreducible over $K(a)$ My attemp: (1) Let deg $\...
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When does $p(x^k)$ have irreducible factor $(x-\sqrt[k]{x_1})(x-\sqrt[k]{x_2})\cdots(x-\sqrt[k]{x_n})$?

Suppose that $p(x)=(x-x_1)(x-x_2)\cdots(x-x_n)$ is a monic irreducible polynomial over $\mathbb{Z}$. When does $p(x^k), k\geq1$ have irreducible factor $(x-\sqrt[k]{x_1})(x-\sqrt[k]{x_2})\cdots(x-\...
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Every rational Polynomial is a product of the content and a primitive Polynomial

If $f\in \mathbb{Q}[X]\setminus \{0\}$ then $\, f=cont(f)\cdot f_1$ with $f_1 \in \mathbb{Z}[X]$ being a primitive Polynomial. Why is that the case?
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Small question about Gauss's lemma (polynomial)

There is a beautiful proof for Gauss's lemma on Wikipedia here. There is just the last bit I don't understand. It says: "This sum contains a term $a_r b_s$ which is not divisible by p (by Euclid's ...
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1answer
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Problem of isomorphism and counting on quotient ring $\mathbb{Z}_{101}[x]/<f(x)>$

Let $f(x)=x^{101}-x$ and $g(x)=x^{101}-x+1$ in ring $\mathbb{Z}_{101}[x]$. 1) Are quotient rings $$\mathbb{Z}_{101}[x]/< f(x) > and ~~\mathbb{Z}_{101}[x]/<g(x)>$$ isomorphic? 2) ...
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Over which fields do there exist irreducible polynomials of every degree?

Let $F$ be a field and $n \geq 2$. Must there exist an irreducible polynomial of degree $n$ in $F[X]$? When $F=\mathbb{Q}$ the answer is certainly "yes," as you can apply Eisenstein's criterion to $...
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3answers
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Showing the irreducibility of $x^8 - 60 x^6 + 1160 x^4 - 7800 x^2 + 8836$ in $\mathbb{Q}[x]$

I would like to show the irreducibility of $x^8 - 60 x^6 + 1160 x^4 - 7800 x^2 + 8836$ and $x^8 - 120 x^6 + 4360 x^4 - 45600 x^2 + 15376$ in $\mathbb{Q}[x]$. In both cases Eisenstein criterion fails. ...
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2answers
50 views

Why is $\frac{1}{x+2} = \sum_k^\infty (-1)^k(1+x)^k \in \mathbb{Z} [[x]]$ not the inverse of $x+2$ in $\mathbb{Z}$?

As also stated in here, a formal power series is a unit in $R[[x]]$ iff it is constant coefficient is a unit in the ring $R$. However, for example, we can find the inverse of $1+x$ by observing that $\...
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0answers
36 views

Generators of $\mathbb{Z}_2[x][x^3+x^2+1]$

I need to find all generators of the field $\mathbb{Z}_2[x][x^3+x^2+1]^*$ The star is defined as follows: $ F[x]^*m(x) = \{ a(x) \in F[x]m(x) | gcd(a(x), m(x))=1\}$ So this means we only look at the ...
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1answer
42 views

How to prove a polynomial in $\mathbb{Z}[x,y]$ is irreducible

I had the following question: Let $f(x,y) = y^5+xy^2+x \in \mathbb{Z}[x,y]$ and let: $I_2 = (f(x,y),x-1,2)$ and $I_3 = (f(x,y),x-1,3)$ be two ideals in $\mathbb{Z}[x,y]$. Prove that $f(x,...
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2answers
74 views

Is $x^4 + 2x^2 - x + 1$ irreducible in $\mathbb Z_7[x]$?

How would one do this? I know since it doesn't have roots it can only be divisible by irreducible polynomials of degree 2. How would I prove that it is or it isn't? There are a lot of polynomials ...
0
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2answers
30 views

Motivation for the method of adjoining roots of polynomials

In Galois theory we learned the standard method of adjoining a root of an irreducible polynomial. More precisely, we saw that if $K$ is a field and $f\in K[x]$ is irreducible then the field $K[x]/(f)$ ...
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4answers
358 views

Irreducible polynomials of degree greater than 4 over finite fields

I want to build a field with p^n elements. I know that this can be done by finding a irreducible (on Z_p) polynomial f of degree n and the result would be the Z_p/f. My question is finding this ...
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1answer
28 views

Irreducible polynomial in the extension k(x)/k(u).

If I consider $k$ a field of prime characteristic (maybe this is not important here...). Consider the field of fractions $k(x)$ and $u \in k(x)$ where $u=\frac{f(x)}{g(x)}$ where $f$ and $g$ are ...
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0answers
37 views

For a line $L$ and an algebraic curve $C$ of an irreducible polynomial, prove $C \cap L$ contains at most d points unless C = L.

Artin Algebra Chapter 11 This has been answered here. My questions are about the solution of Brian Bi: By stronger, does he mean that $C \ne L$ and $f$ is irreducible $\implies l \nmid f?$ If so, ...