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 […]

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 […]

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 […]

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 […]

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 […]

(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 = […]

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 […]

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 […]

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 […]

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.

Intereting Posts

Why do complex functions have a finite radius of convergence?
Generalisation of $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$
Prove that $=$
Inner Product Spaces – Triangle Inequality
The contradiction method used to prove that the square root of a prime is irrational
Show that an entire function that is real only on the real axis has at most one zero, without the argument principle
Counting symmetric unitary matrices with elements of equal magnitude
Understanding matrices as linear transformations & Relationship with Gaussian Elimination and Bézout's Identity
Spectrum of $\mathbb{Z}^\mathbb{N}$
Why is the prime spectrum of a domain irreducible in the Zariski topology
$\frac{1}{e^x-1}$, $\Gamma(s)$, $\zeta(s)$, and $x^{s-1}$
Locus of intersection of two perpendicular normals to an ellipse
solution of ordinary differential equation $x''(t)+e^{t^2} x(t)=0, for t\in$
Prove $\alpha \in\mathbb R$ is irrational, when $\cos(\alpha \pi) = \frac{1}{3}$
Is $\mathbb{Z}/p^\mathbb{N} \mathbb{Z}$ widely studied, does it have an accepted name/notation, and where can I learn more about it?