All Questions
27
questions
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40
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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) =...
- 474
1
vote
0
answers
41
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Minimal polynomial of a complex number $z$ that is also an algebraic integer does not have $z$ as a repeated root [duplicate]
Let $z\in \mathbb{C}$ be an algebraic integer, and let $P\in \mathbb{Z}[X]$ be its minimal polynomial over $\mathbb{Z}$. Prove that $(X-z)^2 \nmid P(X)$.
My attempt:
My first idea was to prove that $P'...
- 1,144
4
votes
2
answers
228
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Help with a finite field exercise. How to find the minimal polynomial of a given root in a given field.
I need a help with this exercise.
(i) Find a primitive root $\beta$ of $\mathbb{F}_2[x]/(x^4+x^3+x^2+x+1)$.
(ii) Find the minimal polynomial $q(x)$ in $\mathbb{F}_2[x]$ of $\beta$.
(iii) Show that $\...
- 391
0
votes
0
answers
36
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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 + \...
2
votes
2
answers
327
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*}...
- 1,242
0
votes
1
answer
172
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Irreducible cubic polynomial in $\mathbb{Q}[x]$ has no roots in $\mathbb{Q}(\sqrt{2}, 5^{1/4})$
I am working on the following problem:
Consider the field extension $F = \mathbb{Q}(\sqrt{2}, 5^{1/4})$ of $\mathbb{Q}$. Let $f(x)$ be an irreducible cubic polynomial in $\mathbb{Q}[x]$. Prove that $...
- 1,503
3
votes
1
answer
109
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Where is the flaw in my reasoning about the number of irreducible degree 9 polynomials over $\mathbb F_2$?
I am aware of the formula for the number of polynomials of degree $n$ over a finite field, and that it gives 56 for this particular case, but I wanted to know why the following train of thought is ...
- 2,683
0
votes
2
answers
831
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What is the Galois group of $f = X^4 - 3X^2+3 \in \mathbb{Q}[X]$?
The polynomial $f$ is an irreducible Eisenstein polynomial with $p = 3$.
Its roots are easy to find using the substitution $Y = X^2$ and then the $abc$-formula: $\{\sqrt{\frac{3}{2}+\frac{\sqrt{-3}}{...
- 1,334
1
vote
2
answers
56
views
A minimal polynomial problem that's bothering me
Let $F$ be a field and $P$ an irreducible polynomial in $F[x]$. If I find a root of $P$ in an extention of $F$, does that make $P$ a minimal polynomial?
user372244
3
votes
3
answers
647
views
Given degree 3 minimal polynomial of $\alpha$, find minimal polynomial of $\frac{\alpha^2 + \alpha}{2}$
I am interested in techniques that you might be able to use to compute the minimal polynomial of basic functions of a root of a minimal polynomial you know. For example $\alpha^2 + \alpha,\; 1 + \...
- 2,683
0
votes
0
answers
22
views
Existence of Irreductible polynomial
Known that $A,B,C$ are coprime in K[X]
if $AB+BC+AC$ and $ABC$ aren't coprime then there exists an IRREDUCIBLE polynomial $D$ that divides them both .
Why is D irreducible?
- 1
1
vote
4
answers
431
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Fastest method to verify $x^4-2x^2+9$ is irreducible so minimal
I was tasked with finding $\min(\mathbb{Q},i+\sqrt[]{2})$
I found the polynomial in $\mathbb{Z}[X]$ to be $X^4-2X^2+9$
I know it must be minimal over $\mathbb{Q}$ since (used brute force) any ...
- 548
1
vote
0
answers
146
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Link between dominant root of a polynomial and its coefficients
My problem is the following:
Let $k$ be a strictly positive integer and let's look at a polynomial $$X^k - c_0 X^{k-1} - \ldots - c_{k-1}$$
that we suppose irreductible over $\mathbb{Z}$, where $c_0, ...
- 151
2
votes
0
answers
88
views
Minimal polynomial of a trigonometric number
I am trying to calculate the minimal polynomials of $h_{1}=-\cos(\pi/n)-\sqrt{\cos(2\pi/n)}$ and $h_{2}=-\cos(\pi/n)+\sqrt{\cos(2\pi/n)}$ when $n$ is odd. I think (and numerical calculations suggest ...
0
votes
1
answer
83
views
How to find the minimal polynomial of a polynomial over a finite field
Let $\mathbb{F}_{p^n}$ be a finite field and $f(x) \in \mathbb{F}_{p^n}[x]$ which is monic and non-constant. My question is:
$i)$ Is there the minimal polynomial $g(x) \in \mathbb{F}_{p}[x] \subset \...
- 773