Questions tagged [irreducible-polynomials]

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

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32 views

Every irreducible component of the zero set of a single polynomial $f$ is the zero set of an irreducible factor of $f$

Let $k$ be a field and is algebraically closed. Let $f\in k[x_{1},\cdots, x_{n}]$ be a polynomial (not necessarily irreducible). As $k[x_{1},\cdots, x_{n}]$ is a UFD, we can write $f=ug_{1}\cdots g_{n}...
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Basic theorem of field extensions

I am studying the the basic theorem of field extensions with Charles C. Pinter's book: A book of Abstract Algebra. The very same theorem that is being discussed here: Example of Basic Theorem of field ...
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Family of irreducible polynomials over $\mathbb{Z}$

Note. In this question I am using the following definition of irreducible polynomial: "Polynomial is irreducible if it cannot be factored into polynomials with coefficients in the same domain ...
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30 views

Minimal form of rational, integer and complex polynomials?

Do all rational polynomial e.g $\frac{a}{b}x^n + \frac{c}{d}x^{n-1} + ... +\frac{e}{f}x + \frac{f}{g} = 0$ have a integer polynomial representation? My thinking is: $\frac{a}{b}x^2 + \frac{c}{d}x + \...
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1answer
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A proof of Eisenstein's criterion and a related proposition in "Lectures on Algebra" by Teiji Takagi.

I am reading "Lectures on Algebra" by Teiji Takagi. The following proposition by Eisenstein is in this book. Let $f(x)=c_0+c_1x+c_2x^2+\dots+c_lx^l$ be a polynomial such that $c_i\in\mathbb{...
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4answers
118 views

Irreducibility of $1+\sqrt{-3}$ in $\mathbb{Z}[\sqrt{-3}]$

We were having a discussion on the irreducibility of $1+\sqrt{-3}$ in $\mathbb{Z}[\sqrt{-3}]$ A classmate pointed out that $1+x$ is irreducible in $\mathbb{Z}[x]$, hence, putting $x=\sqrt{-3} \...
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1answer
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if $ax^2+bx+c$ and $2ax+b$ expressions have a common divisor, then $ax^2+bx+c$ is a full square.

if $ax^2+bx+c$ and $2ax+b$ expressions have a common divisor, then $ax^2+bx+c$ is a full square. I would like to prove this statement. Thank you for any idea.
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2answers
50 views

Irreducible factors of minimal and characteristic polynomial of a endomorphism over a finite dimensional $\mathbb{F}$-vector space [duplicate]

Let $V$ be a finite dimensional $\mathbb{F}$-vector space. Suppose $L:V\to V$ is an endomorphism, whose associated matrix is $A$. Now, denote its characteristic and minimal polynomial by \begin{align*}...
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$f,g$ be irr poly of degree $m$ and $n$. Show that if $\alpha$ is a root of $f$ in some extension of $F$, then $g$ is ireducible in $F(\alpha)[x]$

Question: let $f,g\in F[x]$ be irreducible polynomials of degree $m$ and $n$, respectively, for $(m,n)=1$. Show that if $\alpha$ is a root of $f$ in some extension of $F$, then $g$ is ireducible in $...
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64 views

Prove that a polynomial is irreducible

Suppose that $f(x,1)=ax^3+bx^2+cx+d$ is irreducible polynomial, with integer coefficient and such that $ \gcd(a,b,c,d)=1$, with some root $ \theta$, i want to prove that $g(x,1)=x^3+bx^2+acx+a^2d$ is ...
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1answer
115 views

Quotient ring $\frac{\mathbb{Z}_n[x]}{⟨f(x)^2⟩}$

I know from Chinese remainder theorem that: If $\mathbb{F}$ is a field and suppose that $ f(x)\in\mathbb{F}[x]$ is factored into distinct irreducible factors $f(x)=f_1(x).f_2(x)...f_m(x)$,then we ...
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Irreducibility of $p^{n-1}X^n+pX+1$ over $\mathbb{Q}$ [duplicate]

I am attempting to show that $f(X)=p^{n-1}X^n+pX+1$ are irreducible over $\mathbb{Q}$ for any positive integer $n$ and any prime $p$. At the behest of my teacher, and their hint, I would like to do so ...
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133 views

Is $x^3-ax-(2a+1)$ irreducible over $\Bbb Z$?

Is the cubic polynomial $x^3-ax-(2a+1)$ irreducible over $\mathbb Z$ for all positive integers $a$? one way to prove irreducibility is to use eisenstein's criterion. we want to find prime $p$ st: $p\...
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27 views

Irreducible decomposition of algebraic set via a decomposition of the polynomial

I am trying to understand the proof of the following theorem: Let $f \in \mathbb{C}[x,y]$ and $f = \prod_{i=1}^{r} f_i^{n_i}$ be a decomposition of $f$ into irreducible factors. Then, $V(f) = \cup_i V(...
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Question on why a field extension isn't a splitting field (and on general behavior of field extensions)

In class, we were discussing splitting fields, and I was wondering why the quotient of a polynomial ring by an irreducible polynomial "prioritizes" certain roots in the case that all of a ...
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41 views

Irreducible algebraic variety of dimension $d$ cannot necessarily be given by $n-d$ equations

Let $k$ be an algebraically closed field. For an algebraic set $Y\subset k^n$ it is true that $Y$ is irreducible and of dimension $n-1$ iff $Y=Z(f)$ for some irreducible $f\in k[x_1,\dots,x_n]$. My ...
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How can you find the roots of a polynomial if all the roots are imaginary?

I am trying to find the roots of a 4th-degree polynomial by hand which is actually coming from the derivative of the open-loop transfer function of a control system to find the breakaway points for ...
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1answer
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Discuss $\mathbb R[X]/(aX^2 +bX + c)$ in terms of $\Delta = b^2-4ac$

Discuss $\mathbb R[X]/(aX^2 +bX + c)$ in terms of $\Delta = b^2-4ac$. I've already found that $\mathbb R[X]/(aX^2 +bX + c) \simeq \mathbb{R} \times \mathbb{R}$ if $\Delta > 0$ and $\mathbb R[X]/(...
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Homogenous polynomials and irreducibility

Hi i'm trying to do an exercise of algebra and i'm stuck. Here it goes Let $A$ be a factorial ring, $ F_n$ and $F_{n+1}$ two non-zero polynomials of $A[X_1,\cdots ,X_k](k\in \mathbb{N}^{*})$, ...
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Question about irreducible polynomials in algebraically closed fields

A tipical question in the past exams of a commutative algebra course is to determine if a ring of the form $k[x_1,\dots,x_n]/f(x_1,\dots,x_n)$ is an integral domain, usually with $1\le n \le 3$. ...
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113 views

Eisenstein criterion and Newton polygon

Eisenstein criterion of irreducibility over $\Bbb Q_p$(so ofcourse over $\Bbb Q$) is also proved by using Newton polygon. The proof goes like this, Let $f(x)=a_nx^n+・・・+a_1x+a_0$ be $p$-Eisenstein ...
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How to tell if a multivariable polynomial is irreducible?

If I am given a multivariable polynomial $F(x_1,...,x_n)$ with real coefficients, how do I tell if it is irreducible? Similarly, I would like to be able to do this for polynomials with complex ...
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1answer
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Is $(p^{\frac{p^2+1}{p^5-1}}-x^{p^3})^{p^2}+p^{p^2+1}x-p^{p^2+\frac{p^2+1}{p^5-1}} $ eisenstein?

I could not decide whether the following polynomial is $$f(x)=(p^{\frac{p^2+1}{p^5-1}}-x^{p^3})^{p^2}+p^{p^2+1}x-p^{p^2+\frac{p^2+1}{p^5-1}}$$ an eisenstein polynomial? If I write simply \begin{...
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1answer
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Irreducibility of $x^n+px+p^2(n≧3)$ and newton polygon

It is well known that If $f$ is irreducible polynomial over $\Bbb Q_p$, then there is only one slope, i.e. the newton polygon of $f$ consists of a single segment. Now, newton polygon of $x^n+px+p^2(...
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1answer
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Rupture field of an irreducible polynomial over a finite field equals its splitting field

Let $k= \mathbb{F}_{q}$ a finite field and $P \in k[X]$ an irreducible polynomial. Show that its rupture field is also its splitting field. My take : Let $K$ be the splitting field of P. By the ...
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1answer
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Proving $1+x+\cdots +x^{p-2}+x^{p-1}$ is irreducible for prime $p$ [duplicate]

Prove that if $p$ is a prime number, the polynomial $f(x)=1+x+\cdots +x^{p-2}+x^{p-1}$ is irreducible in $\mathbb{Z}[x]$. I tried using Eisenstein Criterion and the Rational Root Theorem, also, I know ...
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1answer
73 views

How to prove that the polynomial $x^2 + x + 1$ is irreducible over $\mathbb Q(\sqrt[3]{2})$?

I am trying to prove that the polynomial $x^2 + x + 1$ is irreducible over $\mathbb Q(\sqrt[3]{2}).$ This is my guess: $x^2 + x + 1$ is reducible over $\mathbb Q(\sqrt[3]{2})$ iff $\xi \in \mathbb Q(\...
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1answer
65 views

Prove the irreducibility of $P(x)$ which satisfies: $xP(x-1)=(x-2022)P(x)+2022$

Prove the irreducibility in $\mathbb{Z}[x]$ of $P(x)$ which satisfies: $xP(x-1)=(x-2022)P(x)+2022, \forall x\in \mathbb{R}$ My attempts I thought of substituting $x$ for $0$ and $x$ for $2022$, but ...
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1answer
75 views

Factor $X^7 − 1$ into irreducibles in $\mathbb{Z}_{127}[X]$

Problem: Factor $X^7 − 1$ into irreducibles in $\mathbb{Z}_{127} [X]$ I'm having a lot of trouble understanding how to approach this, and this course topic in general. Any help (or even better, a ...
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Romania TST 1995 -irreducible Polynomail

Question is: Let $f$ be an irreducible (in $Z[x]$) monic polynomial with integer coefficients and of odd degree greater than $1$. Suppose that the modules of the roots of $f$ are greater than $1$ and ...
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Checking if given polynomials are units in $\mathbb{Z}_7[x]$ [duplicate]

So, I was doing an algebra exercise related to $\gcd$'s and $PID$'s and I need to check if some polynomials in $\mathbb{Z}_7[x]$ are units. The polynomials are the following: \begin{equation*} g = x^2+...
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Separable and irreducible polynomial over $F$

Let $f(x) \in F[X]$ a polynomial with degree 5 and Galois Group $S_5$. Show that $f(x)$ is separable and irreducible over $F$. For irreducibility, I tried by contradiction $$ f(x) = g(x) * h(x) $$ ...
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Showing $f(x) = x^4 - 6x^2 - 8x + 3$ is irreducible in $\mathbb{Q}[x]$ [duplicate]

I wish to determine that $f(x) = x^4 - 6x^2 - 8x + 3$ is irreducible in $\mathbb{Q}[x]$ by substituting $x + m$ for $x$ and then choosing $m$ appropriately. $\textit{Work so far:}$ By rational root ...
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1answer
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Invertible elements of $K[x]/\langle p(x)\rangle$ when $p(x)$ is irreducible

I want to find the inverse of $f(x)=ax^{2}+bx+c$ in $L=K[x]/\langle p(x)\rangle$ when $p(x)$ is an irreducible polynomial in $K[x]$ with degree $3$. I know elements of $L$ are like $f(x)$ such that $...
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1answer
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Infinite irreducible polynomial over Q[x] using Eisenstein

Like the title describes, I know that over Q for each number n ≥ 1, one can easily construct infinitely many irreducible polynomials of degree n. But I want to prove using Eisenstein's criterion this ...
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1answer
57 views

Existence of irreducible Polynomials in finite fields

I want to prove that given K a finite field and n>0, exists an irreducible polynomial f ∈ K[x] of degree n. I am taking into account this explanation given in Field Theory by Roman: But I don't ...
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1answer
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Factoring $t^{q^r}-t$ over the finite field $\mathbb{F}_q$

This might be a very short-lived question. Let $F$ be a finite field of order $q$ ($q$ is a prime power). It is well known that the polynomial$$t^{q^d}-t$$factors into the product of all irreducible ...
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Number of roots of irreducible polynomials in extension fields

Let $k$ be a field and $p$ be an irreducible polynomial of degree $n$ in $k[x]$. Let $E$ be a field extension of $k$. Can anything be said about the number of roots of $p$ that are present in $E$? I ...
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1answer
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Whether $\mathbb{Q}[x]/(f)$ and $\mathbb{Q}[x]/(g)$ are isomorphic, with $f,g$ irreducible cubic polynomials [closed]

I' trying to check if the claim in the title is true, and my argument is: If they are isomorphic, then they are isomorphic after a base change to $\mathbb{F}_p$, then I found a prime number $p$ such ...
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1answer
52 views

Irreducibility of a polynomial in $\mathbb{Q}[x,y]$ [closed]

Let $x^4+x^3y+x^2y^2+xy^3+y^4 \in \mathbb{Q}[x,y]$ be a primitive polynomial for which we have to investigate if it is irreducible. My idea is: since $(y-1) \in \mathbb{Q}[x,y]$ is a prime ideal such ...
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1answer
119 views

show that:this $f(x)=x^n-k$ is irreducible in $\mathbb Z[x]$

Let $n$ be postive integer, and $k$ is integer. If there exists a prime number $q$,and postive integer $a$, such $v_{q}(k)=a$,and $(a,n)=1$.show that: $$f(x)=x^n-k$$ is irreducible in $\mathbb Z[x].$ ...
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56 views

Finding minimal polynomial of $\sqrt3 + i \sqrt 7$ over $\Bbb Q$

$x=\sqrt 3 + i \sqrt 7$ I have come to the following polynomial: $x^4 + 8x^2+100=0$. By Eisenstein criterion, this pol is not irreducible in $\Bbb Q[x]$, but I don't know how to factor it further? ...
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24 views

Shifted polynomial identical (except for constant term)?

I have a function $f(x)$ and $g(x)$ such that when the function is shifted by some integer value $m$ and constant term $c$, it is possible to get $f(x+m)=g(x)+c$. I was wondering what this property ...
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61 views

In Mod p Irreducibility Test for Z[t] Need p be Prime?

In descriptions of the mod $p$ irreducibility test it is always stated $p$ is prime. However is that restriction needed, since for any positive integer $m$ we have : If $m \nmid \mathrm{LC}(f)$ and $f ...
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0answers
43 views

Irreducibility of $x_1^2+x_2^2+...+x_n^2$ in $\mathbb{C}[x_1,...,x_n]$ [duplicate]

I wanted to know whether the polynomial function $x_1^2+...+x_n^2 \in \mathbb{C}[x_1,...,x_n]$ is irreducible in $\mathbb{C}[x_1,...,x_n]$ for $3 \leq n$ ?
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4answers
107 views

How to prove $X^{4}+X^{3}+X^{2}+X+1$ is irreductible in $\mathbb{F}_{2}$

How to prove $X^{4}+X^{3}+X^{2}+X+1$ is irreducible over $\mathbb{F}_{2}$. The main 2 "weapons" I have at my disposal is the Eisenstein criteria and reduction criteria but neither seem to ...
2
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1answer
119 views

Irreducible polynomials in $\mathbb Z[X]$ with no primes in the image.

It seems intuitive that if an irreducible polynomial $q\in\mathbb Z[X]$ has no primes in it's image, i.e. $n\in\mathbb Z\implies q(n)\notin \mathbb P$, then there is a non unit integer $m\in\mathbb Z$ ...
4
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1answer
114 views

Minimal polynomial of $x=\sqrt{2}+i\sqrt{3}$

I was asked to calculate the minimal polynomial of $x=\sqrt{2}+i\sqrt{3}$ over the fields \begin{align*} K_1 = \mathbb{Q}, \quad K_2 = \mathbb{Q}(\sqrt{2}), \quad K_3 = \mathbb{Q}(i\sqrt{3}), \...
6
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1answer
193 views

Can we prove those conjectures about a family of polynomials?

Let $\ a,b\ $ integers with $\ 0<a<b\ $. Consider the following family of polynomials : $$f(x)=x^b+x^a+1$$ $$g(x)=x^b+x^a-1$$ $$h(x)=x^b-x^a+1$$ $$i(x)=x^b-x^a-1$$ The problem is about which of ...
3
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1answer
84 views

Divisibility in $\mathbb{Q}[x]$

Let $F(x),G(x) \in \mathbb{Z}[x]$ with $G(x)$ being irreducible in $\mathbb{Z}[x]$. (By irreducible, we mean that if $G(x)=A(x)B(x)$ for $A(x),B(x) \in \mathbb{Z}[x]$, then either $A(x)$ or $B(x)$ is ...

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