Questions tagged [minimal-polynomials]

Question 1

Let $V$ be an $\mathbb{F}$-vector space, over an algebraically closed $\mathbb{F}$, without any topology and possibly infinitely dimensional. If $T\in L(V)$ admits a minimal polynomial $p_T=\prod_\...

Question 2

Simplify into reduced form the expression $$\frac{1+2α+3α^2} {(1+α)^{15}} $$ , where the minimal polynomial of $α ∈ \mathbb{C}$ over $\mathbb{Q}$ is $x^3 + x + 1$. My Attempt: First, we can write $\...

Question 3

Let $\mathbb{F}$ be a sub-field of $\mathbb{C}$, $V$ be a finite dimensional $\mathbb{F}$-vector space, and $T\in \operatorname{End}(V)$ be semi-simple. Show that for all $f\in \mathbb{F}[x]$, $f(T)$ ...

Question 4

Let $f(X) = a_nX^n+\cdots+a_0 \in \mathbb{Z}[X]$ be a polynomial with roots $\alpha_1,\dots,\alpha_n$. The Mahler measure of $f$ is defined to be $$ M(f) = |a_n|\prod_{i=1}^{n}\max\{1,|\alpha_i|\}. $$ ...

Question 5

I am trying to use polynomial roots to approximate this ratio $R(p)$: $$R(p)=\Re\left(\frac{\underset{k\to \infty}{\text{lim}}\left(\left(H_k^{(s)}\right)^{1/p}+\left(\frac{k^{1-s}}{s-1}\right)^{1/p}\...

Question 6

Problem: Find the minimal polynomial of $\sqrt{3}+\sqrt{5}$ over $\mathbb{Q}$ and show that it is an irreducible polynomial in $\mathbb{Z}[t]$ which becomes reducible modulo any prime $p$. I found ...

Question 7

Let $K=\mathbb{Q}(\theta)$ be a number field where $\theta$ is algebraic with minimal polynomial $f(x) \in \mathbb{Z}[x]$. Also, let $\mathcal{o}$ denote its ring of integers, and consider a rational ...

Question 8

Let $k$ be a field and $V$ an $n$-dim $k$-vector space. Then, for every nonzero $A\in L(V)$, the evaluation map $EV_A:k[X]\to L(V)$, by $f(x)\mapsto f(A)$ is a unital ring homomorphism. The kernel of ...

Question 9

Let $F= \mathbb{F}_{3}(a)$ a field with $a$ verifying the equation $a^{3}+a^{2}-1 = 0$. Determine the cardinality of $F$. Determine the degree of $Irr(a^{2}, \mathbb{F}_{3})$ Compute $Irr(a^{2}, \...

Question 10

Let $F$ be the splitting field of $f = x^{2}+x+1 \in \mathbb{F}_{5}[x]$ and let $a \in F$ be a root of $f$. Prove that the polynomial $g = Irr(a+1, \mathbb{F}_{5})$ verifies $g(a) = 3a$. We first show ...

Question 11

Let $A \subseteq B$ be two GCD domains with $1$. Assume that the extension $A \subseteq B$ is algebraic, not necessarily integral. Notation: For $w \in B$ denote its minimal polynmial over $A$ by $m_w$...

Question 12

I’m studying the algebraic structure of $\cos(2\pi/13)$, which is known to be an algebraic number of degree $6$ over $\mathbb{Q}$. It’s also known from Gaussian period theory that: $$ \sqrt{13} = -2 \...

Question 13

There is a result that states that if "$V$ is finite-dimensional, $T \in L(V)$, and $U$ is a subspace of $V$ that is invariant under $T$, then the minimal polynomial of $T$ is a polynomial ...

Question 14

Problem: Find the minimal polynomial of $t=\sqrt{13+6\sqrt{2}}\in\mathbb{R}$ over $\mathbb{Q}$. I found that $f(x)=x^4-26x^2+97\in\mathbb{Q}[x]$ satisfies $f(t)=0$. Now we need to show that $f$ is ...

Question 15

Basic fact: for an $n$-dim space $V$ and nonzero $A\in L(V)$, let $p_A$ be its minimal polynomial, then $\deg(p_A)\leq n$. However, I’ve only seen a proof based on induction, and I can’t seem to ...

Stack Exchange Network

Questions tagged [minimal-polynomials]

Infinite dimensional operators admiting a minimal polynomial still decomposes the space into invariant subspaces?

Simplify into reduced form

Polynomials of semi-simple linear operator is semi-simple

The Mahler measure of a integral polynomial is an algebraic integer

Using the analytic continuation of the Riemann zeta function to approximate some polynomial roots.

Show that the minimal polynomial of $\sqrt{3}+\sqrt{5}$ over $\mathbb{Q}$ is irreducible in $\mathbb{Z}[t]$ but becomes reducible modulo any prime $p$

Proving $\mathbb{Z}[x]/(p,f(x))\cong \mathcal{o}_p$ for most primes $p$ where $f(x)$ is a minimal polynomial [closed]

Matrix polynomials and quotient field

Degree of the minimal polynomial over a finite field.

How can I find the minimal polynomial over a finite field in a different way?

Leading coefficients of minimal polynomials of $u,v$ and $u+\lambda v$

Is there a minimal-degree integer polynomial $f(x)$ such that $f(\cos(2\pi/13)) = \sqrt{13}$?

Minimal polynomial of $T$ divided by minimal polynomial of $T|_U$ gives a a contradiction?

Find the minimal polynomial of $\sqrt{13+6\sqrt{2}}\in\mathbb{R}$ over $\mathbb{Q}$

Degree of minimal polynomial of a square matrix no larger than dimension

Hot Network Questions