Articles of minimal polynomials

Find the Irreducible Polynomial of This Element over $\mathbb Q$

Let $K = \mathbb Q(a)$ where $a$ is a root of $x^3 – x – 1$. Find the irreducible polynomial of $c = 1 + a^2$ over $\mathbb Q$. I found the answer by brute force. Write: \begin{align} a^3 & = a + 1 \\ a^4 & = a^2 + a \\ a^5 = a^3 […]

Compute the minimal polynomial of $u$ over $\Bbb{Q}$ without using cyclotomic polynomials

Let $k\in \Bbb Z\setminus 7\Bbb Z$ and $a_k=\frac{2k\pi}{7}$. Compute the minimal polynomial of $u=$ over $\Bbb{Q}$ The “natural” (in my opinion) way to solve this problem, requires the use of $\Phi_7(t)=t^6+t^5+t^4+t^3+t^2+t+1$, the seventh cyclotomic polynomial, since it has the complex number $e^{2k\pi/7}$ as root. Then, noticing that $$u=e^{2k\pi i/7}+e^{-2k\pi i/7}$$ and dividing $\Phi_7(t)$ by $t^3$, […]

Minimal Polynomial of Inverse

Suppose $T$ has minimal polynomial $x^m+a_{m-1}x^{m-1}+…+a_1x+a_0$ and $T$ is invertible (hence $a_0\not=0$). Is it true that the minimal polynomial of $T^{-1}$ is $\frac{1}{a_0}(1+a_{m-1}x+…+a_1x^{m-1})+x^m$? My thought was that since $T^m+a_{m-1}T^{m-1}+…+a_1T+a_0I=0$, we have $I+a_{m-1}T^{-1}+…+a_1T^{m-1}+a_0T^{-m}=0\implies\frac{1}{a_0}(I+a_{m-1}T^{-1}+…+a_1T^{m-1})+T^{-m}=0$. So the minimal polynomial of $T^{-1}$ divides $\frac{1}{a_0}(1+a_{m-1}x+…+a_1x^{m-1})+x^m$ and so (degree of minimal polynomial of $T^{-1}$)$\leq m=$(degree of minimal polynomial of $T$). On […]

Minimal polynomial of $\sqrt{5}+\sqrt 2$ over $\mathbb{Q}(\sqrt{5})$

Find the minimal polynomial of $\sqrt[3]{5}+\sqrt{2}$ over $\mathbb{Q}(\sqrt[3]{5})$ Attempt: Let $u:=\sqrt[3]{5}+\sqrt{2}\\ u-\sqrt[3]{5}=\sqrt 2\\ (u-\sqrt[3]{5})^2=2\\ \boxed{u^2-2\sqrt[3]{5}u+5^{2/3}-2=0}$ Is this correct? the “over $\mathbb{Q}(\sqrt[3]{5})$” confuses me

Find eigenspace for eigenvalues of a linear transformation $T(M)=-2M^t+M$

Let $V$ be the matrix space $4 \times 4$ over $\Bbb R$. $T: V \to V$ is a linear transformation defined by: $$T(M)=-2M^t+M$$ for all $M \in V$. Find the minimal polynomial of T. For every eigenvalue $\lambda$ of $T$, find the eigenspace $V_\lambda$ and calculate its dimension. Find $T$’s characteristic polynomial. I have solved […]

Minimal polynomial of $T(A) = A^t – A$

As said in the title , I need to find the min polynomial of that linear transform. The matrices are $M_n(\mathbb{C})$. I’ve figured out that $T^2 = 2A – 2A^t$ , so a polynomial $p(t) = t^2 + 2t$ works so $p(T) = 0$. Now $p(t)$ breaks to $t(t+2)$ but non of them kills T. […]

Minimal polynomial of $A := \left(\begin{smallmatrix} 7 & -2 & 1 \\ -2 & 10 & -2 \\ 1 & -2 & 7 \end{smallmatrix}\right)$

I realised an error in my previous attempt, before writing the question, but think this can help some guys to solve their own multipolynomial problems, if you want to keep it (so they can find it via searchfunction) tell me otherwise I will just delete it all.. Minimal polynomial of $A := \begin{pmatrix} 7 & […]

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 […]