Questions tagged [minimal-polynomials]

This is the lowest order monic polynomial satisfied by an object, such as a matrix or an algebraic element over a field.

For instance, $\sqrt2$ is an algebraic number, that is, it's a root of a non-zero polynomial with rational coefficients. Its minimal polynomial is $x^2-2$, since $\sqrt2$ is a root of this (monic) polynomial, and it is not a root of a non-zero polynomial with rational coefficients of a smaller degree.

1146 questions
61
votes
7 answers

Do matrices $ AB $ and $ BA $ have the same minimal and characteristic polynomials?

Let $ A, B $ be two square matrices of order $n$. Do $ AB $ and $ BA $ have same minimal and characteristic polynomials? I have a proof only if $ A$ or $ B $ is invertible. Is it true for all cases?
43
votes
3 answers

When are minimal and characteristic polynomials the same?

Assume that we are working over a complex space $W$ of dimension $n$. When would an operator on this space have the same characteristic and minimal polynomial? I think the easy case is when the operator has $n$ distinct eigenvalues, but what…
39
votes
2 answers

Minimal polynomials and characteristic polynomials

I am trying to understand the similarities and differences between the minimal polynomial and characteristic polynomial of Matrices. When are the minimal polynomial and characteristic polynomial the same When are they different What conditions…
32
votes
5 answers

The characteristic and minimal polynomial of a companion matrix

The companion matrix of a monic polynomial $f \in \mathbb F\left[x\right]$ in $1$ variable $x$ over a field $\mathbb F$ plays an important role in understanding the structure of finite dimensional $\mathbb F[x]$-modules. It is an important fact that…
30
votes
3 answers

$f(x) $ be the minimal polynomial of $a$ (algebraic element) over $\mathbb Q$ , let $b=f'(a) \in \mathbb Q(a)$ , then is $\mathbb Q(a)=\mathbb Q(b)$?

Let $a \in \mathbb C$ be algebraic over $\mathbb Q$ , let $f(x) \in \mathbb Q[x]$ be the minimal polynomial of $a$ over $\mathbb Q$ , let $b=f'(a) \in \mathbb Q(a)=\mathbb Q[a]$ , then is it true that $\mathbb Q(a)=\mathbb Q(b)$ ?
user228168
27
votes
4 answers

Does every linear operator have a minimal polynomial?

I know that a linear operator $T$ defined on a finite-dimensional vector space has a minimal polynomial since, by Caley-Hamilton, $g(T)=0$, where $g$ is the characteristic polynomial. Is there a linear operator defined on an infinite-dimensional…
Wylnorr
  • 533
  • 4
  • 13
23
votes
1 answer

What comes after $\cos\left(\tfrac{2\pi}{7}\right)^{1/3}+\cos\left(\tfrac{4\pi}{7}\right)^{1/3}+\cos\left(\tfrac{6\pi}{7}\right)^{1/3}$?

We have, $$\big(\cos(\tfrac{2\pi}{5})^{1/2}+(-\cos(\tfrac{4\pi}{5}))^{1/2}\big)^2 = \tfrac{1}{2}\left(\tfrac{-1+\sqrt{5}}{2}\right)^3\tag{1}$$ $$\big(\cos(\tfrac{2\pi}{7})^{1/3}+\cos(\tfrac{4\pi}{7})^{1/3}+\cos(\tfrac{6\pi}{7})^{1/3}\big)^3 =…
19
votes
5 answers

Prove $f(x)=x^8-24 x^6+144 x^4-288 x^2+144$ is irreducible over $\mathbb{Q}$

How to prove $f(x)=x^8-24 x^6+144 x^4-288 x^2+144$ is irreducible over $\mathbb{Q}$? I tried Eisenstein criteria on $f(x+n)$ with $n$ ranging from $-10$ to $10$. None can be applied. I tried factoring over mod $p$ for primes up to $1223$. $f(x)$…
19
votes
4 answers

Finding the minimal polynomial of $\sqrt 2 + \sqrt[3] 2$ over $\mathbb Q$.

I have to find the minimal polynomial of $\sqrt 2 + \sqrt[3] 2$ over $\mathbb Q$. The suggested way of doing it is to prove that $\mathbb Q[\sqrt 2 + \sqrt[3] 2]=\mathbb Q[\sqrt 2,\sqrt[3] 2]$ first. I can prove that. It's enough to prove that…
user23211
17
votes
3 answers

Find the minimal polynomial of $\sqrt2 + \sqrt3 $ over $\mathbb Q$

I have no idea how to do this. To find the minimal polynomial of say $\sqrt2 + \sqrt3$, we need to find the monic polynomial $p \in \mathbb Q$ (correct if I am wrong but monic polynomial is when the coefficient of the highest degree term is $1$) of…
16
votes
3 answers

What are some meaningful connections between the minimal polynomial and other concepts in linear algebra?

I’ve found that the most effective way for me to deeply grasp mathematical concepts is to connect them to as many other concepts as I can. Unfortunately, I’m seeing neither the importance nor the relevance of the minimal polynomial at all. Are there…
16
votes
2 answers

Minimal polynomial of product, sum, etc., of two algebraic numbers

The standard proof, apparently due to Dedekind, that algebraic numbers form a field is quick and slick; it uses the fact that $[F(\alpha) : F]$ is finite iff $\alpha$ is algebraic, and entirely avoids the (to me, essential) issue that algebraic…
15
votes
3 answers

Roots of minimal and characteristic polynomials

Why is it that for matrix $A \in M_n(\mathbb{C})$ the characteristic polynomial $\chi_A(t)$ and the minimal polynomial $\mu_A(t)$ have the same roots? Since $\chi_A(t) = \mu_A(t) \cdot p(t)$ it should be easy to follow, that $\chi_A(t)$ has roots…
14
votes
1 answer

What is the exact value of the radius in the Six Disks Problem?

The disk covering problem: Find the smallest radius $r(n)$ required for $n$ equal disks to completely cover the unit disk. For $n=5,6$, the best layouts are, $\hskip2.2in$ $\hskip2.2in$ with $r(5) \approx 0.609,\; r(6) \approx 0.5559$, and $r(5)$ as…
14
votes
2 answers

Are two matrices having the same characteristic and minimal polynomial always similar?

If it is not true, can you provide a counter-example?
xzhu
  • 3,933
  • 1
  • 26
  • 50
1
2 3
76 77