Articles of minimal polynomials

If $F(\alpha)=F(\beta)$, must $\alpha$ and $\beta$ have the same minimal polynomial?

Let’s consider a field $F$ and $\alpha,\beta\in\overline{F}-F$ (where $\overline{F}$ is an algebraic closure). If $F(\alpha)=F(\beta)$, is it true that $\alpha$, $\beta$ have the same minimal polynomial over $F$? For reference, I explain my way. (From here forward, all field isomorphisms are the identity on $F$.) There exist polynomials $f,g\in F[x]$ such that $f(\beta)=\alpha$, $g(\alpha)=\beta$. Let […]

Can every $T$-stable subspace be realised as the kernel of another linear operator that commutes with$~T$?

This is inspired by the question Is Every Invariant Subspace the Kernel of an polynomial applied in the operator?, where “invariant subspace” and “polynomial in” are relative to a given linear operator$~T$ on a finite dimensional vectors space. The answer to that question is a simple “no”, because of simple examples like scalar operators, which […]

Minimal polynomial of an algebraic number expressed in terms of another algebraic number

I am working on the following exercise: Let $\alpha \in \mathbb{C}$ be a root of the polynomial $f(X) = X^4 – 3X – 5$. Prove that $f$ is irreducible in $\mathbb{Q}[X]$. Find the minimal polynomial of $2\alpha – 3$ over $\mathbb{Q}$. Find the minimal polynomial of $\alpha^2$ over $\mathbb{Q}$. Here are my thoughts: I am […]

Hoffman “Linear Algebra”: why need such a long proof?

I’m reading “Linear Algebra” by Kenneth Hoffman and Ray Kunze. I don’t quite understand why there’s a long proof in $\S$6.4 Theorem 6. First the triangular matrix is defined: An $n\times n$ matrix $A$ is called triangular if $A_{ij}=0$ whenever $i>j$ or if $A_{ij}=0$ whenever $i<j$. Then defined triangulable: The linear operator $T$ is called […]

Minimal polynomial for an invertible matrix and its determinant

So here’s one that I can’t quite crack: Let $A\in M_n(\mathbb{F})$ be an invertible matrix with integer eigenvalues. Its minimal polynomial is $m_A(\lambda)=\lambda^k+b_{k-1}\lambda^{k-1}+…+b_0$. Prove that $detA$ is divisible by $b_0$. I’m afraid I don’t quite see the connection between the minimal polynomial and the determinant. All I know is that for an invertible matrix $b_0 […]

Transforming solvable equations to the de Moivre analogues

(This question was inspired by this post.) I. Quintic Given a solvable quintic $$F(x)=0\tag1$$ the problem is to transform it to the de Moivre form $$y^5+5ay^3+5a^2y+b=0\tag2$$ using only a third-degree Tschirnhausen transformation $$y=x^3+c_1x^2+c_2x+c_3\tag3$$ Eliminating $x$ between $(1)$ and $(3)$ yields the quintic $$y^5+d_1y^4+d_2y^3+d_3y^2+d_4y+d_5=0.\tag4$$ The three unknowns $c_i$ allow us to solve the system $$d_1 = […]

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 any significant connections between the minimal polynomial and linear algebra? Particularly regarding […]

Minimal polynomial of the operator $T:V\oplus W\to V\oplus W$

Let $V$ and $W$ be two finite dimensional vector spaces over $\Bbb R$ and let $T_{1}:V\to V$ and $T_{2}:W\to W$ be two linear transformations whose minimal polynomials are given by $f_{1}(x)=x^{3}+x^{2}+x+1$ and $f_{2}(x)=x^{4}-x^{2}-2$. Let, $T:V\oplus W\to V\oplus W$ be the linear transformation defined by $T(v,w)=(T_{1}(v),T_{2}(w))$ for $(v,w)\in V\oplus W$ and let $f(x)$ be the minimal […]

Uniqueness of minimal polynomial: $f(x)$ divides $g(x)$

so I am trying to show that $f(x)$ divides $g(x)$ for all polynomials $g(x)$ satisfying that $g(A)=0$ where $f(x)$ is the minimal polynomial of a square matrix $A$. I know from my professor that $f(x)$ is the minimal polynomial of $A$ so $f(A)=0$ so then $g(A)=0$. Therefore, $f(x)$ divides $g(x)$, i.e. $g(x)=f(x)h(x)$ where $h(x)$ is […]

Minimal polynomial of diagonalizable matrix

Prove that a matrix $A$ over $\mathbb{C}$ is diagonalizable if and only if its minimal polynomial’s roots are all of algebraic multiplicity one.