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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.

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Finding Minimal polynomial from Characteristic polynomial

If $T$ is a linear operator with characteristic polynomial $(x^2-1)^6$ such that $\mathrm{rank}(T-I) = 9$, $\mathrm{rank}(T-I)^2 = 7$, $\mathrm{rank}(T +I) = 10$ and $\mathrm{rank}(T +I)^2 = 9$, find ...
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Finding the minimal polynomial of A with its Characteristic polynomial and dimensions of eigenspaces given.

Let $A$ be a $6\times 6$ complex matrix with Characteristic Polynomial, $c_A(x) = (x^2+1)^3$, $\dim E(i) = 2$ and $\dim E(-i) = 1$. Find the minimal polynomial of $A$.
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Galois theory: Gauss-Wantzel theorem, proof explanation

I am using Ian Stewart Galois theory book and it says that for $A = $ primitive $p^2$ root of unity $A$ has min poly of $m(t)= 1+ t^p +.....+t^{p(p-1)}$ and so $p(p-1)$ is a power of two. why is ...
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Prove that $\frac{Y}{X}$ is irreducible mod $Y^2-X^2(X+1)$

If we write $\frac{Y}{X} = \frac{F}{G}$, do we necessarily have $X$ divide $G$ and $Y$ divide $F$? If this can shed some light, I'm trying to find the set of poles of $\frac{Y}{X}$ mod $Y^2-X^2(X+1)$,...
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Cyclotomic cosets and minimal polynomials

Let $\mathbb{F}_{p^m}$ be a field and let $\alpha \in \mathbb{F}_{p^m}$. Let $M^{(i)}$ be the minimal polynomial of $\alpha^i$. Then I know that $M^{(i)}(x) = \prod_{j \in C_s} (x - \alpha^j)$, where $...
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Finding the minimal polynomial for each algebraic element over $\mathbb{Q}$.

Can someone check whether the following are the correct minimal polynomials for each root? For root $\sqrt{3}+\sqrt[3]{5}$, I got $p(x)=x^{6}-9x^{4}-10x^{3}+27x^{2}-90x - 27$. For root $\cos\theta +...
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Finite Fields, Field Extensions, and Minimal Polynomials

Let $E$ be a field extension of $F$ and let $\alpha \in E$. Define $\phi_{\alpha}:F[x]\to F(\alpha)$ by $\phi_{\alpha}(f(x))=f(\alpha)$. Why is that the kernel of $\phi_{\alpha}$ is the principal ...
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Minimal polynomial and cyclic subspace.

I'm lost with minimal polynomials. I have to prove that the degree of the minimal polynomial $m_{x_0}(x)$ such that cancels a vector $x_0 \in V$ is equal to the dimension of the cyclic subspace ...
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Is there any easy method to find the minimal polynomial of this matrix?

Consider $$A = \begin{bmatrix} 0 &4&1&-2\\-1&4&0&-1\\0&0&1&0 \\-1&3&0&0 \end{bmatrix}$$ Find the minimal polynomial of $A$ . Is there any easy/...
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Characteristic polynomial modulo 12

Consider the vector space $V =\left\{a_0+a_1x+\cdots+a_{11}x^{11},\;a_i\in\mathbb{R}\right\}$. Define a linear operator $A$ on $V$ by $A(x^i) = x^{i+4}$ where $i + 4$ is taken modulo $12$. Find $(...
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Prove that the nullspaces of linear operators acting on themselves are eventually equal

For a linear operator $T$ on a finite-dimensional vector space $V$ such that $dim(V)=n$, prove that $\exists k \leq n$ such that $N(T^k)=N(T^{k+1})$. This is one of those problems where I believe it ...
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minimal polynomial of $\sin (2\pi/7)$ over $\mathbb Q(\sqrt 7)$

How to find the minimal polynomial of $\sin (2\pi/7)$ over $\mathbb Q(\sqrt 7)$ ? And the minimal polynomial of $\sin (2\pi/11)$ over $\mathbb Q(\sqrt {11})$ ? I know that the minimal polynomial of $...
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Degree of field extensions $\mathbb Q(\alpha)/\mathbb Q$ and $\mathbb Q(\alpha^{12})/\mathbb Q$, where $\alpha = 2^{1/3}+3^{1/4}$

How do I find the degree of the following field extensions $\mathbb Q(\alpha)/\mathbb Q$ and $\mathbb Q(\alpha^{12})/\mathbb Q$, where $\alpha = 2^{1/3}+3^{1/4}$ ? Note : Since $\mathbb Q(\alpha) \...
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minimal polynomial of $(2^{1/3}+3^{1/4})^{12}$ over $\mathbb Q$ [closed]

Is there an elegant way to find the minimal polynomial of $(2^{1/3}+3^{1/4})^{12}$ over $\mathbb Q$ ?
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Basis independent proof that $K[X]/p_{_l}\cong K(l)$

For an algebraic field extension $[L:K]$ with $l\in L$ and $p_{_l}\in K[X]$ the minimal polynomial of $l$, is there a basis independent proof that $K[X]/p_{_l}\cong K(l)$? I know a proof that uses $\{...
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Finding the degree of an algebraic field extension

Let $K(\alpha)/K$ be a field extension of degree 4 such that $\alpha^2$ is not a root of the minimal polynomial of $\alpha$ over $K$. Find the degree of $K(\alpha^2)/K$. So far I've been able to show ...
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Common roots of irreducible polynomials

Is it possible for two distinct irreducible polynomials with integer coefficients to have a root in common? In other words, is it possible that a root is shared by some two distinct irreducible ...
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1answer
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Degree of the field of rational numbers extended by a complex number

So I want to calculate the degree of the following fields over the field of the rational numbers: $$\mathbb{Q} \left(e^{\frac{2\pi i}{3}}\right),$$ $$\mathbb {Q} \left(\sqrt{2},\sqrt{1+i}\right)$$ I ...
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The characteristic polynomial of S is $p(z)=(z−1)^2(z−2)^2$. What is the minimal polynomial of S?

Suppose $S∈L(C^4)$ and $B$ is a basis for $C^4$ for which $M(S,B)$ = \begin{bmatrix}2&0&-1/2&1/2\\0&2&1/2&-1/2\\1/2&-1/2&1&1\\-1/2&1/2&1&1\end{bmatrix} ...
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Minimal polynomial of operator with all eigenvalues are 1

I'm trying to understand proof of this lemma. I have next questions: 1)If $\tau$ acts as the identity on the subspace $\mathbb{R}\alpha$ and on the $E/\mathbb{R}\alpha$ doesn't it mean that $\tau$ ...
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Prove that the degree of the minimal polynomial over $\mathbb{Q}$ is a power of $2$

$\textbf{Question.}$ Let $n$ a positive integer and let be $K = \mathbb{Q}[a_1, \cdots, a_n]$ where $a_i$ is such that $a_i^2 \in \mathbb{Q}$ for each $i = 1, \cdots, n$. Given $b \in K$, prove that ...
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invertible matrix and determinant

Assume $(A + I_n)^m = 0$. Prove that $A$ is invertible and find $\det(A)$. I started by binomial expansion, and set it equal zero. is that correct? what would be the best approach?
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Finding the incorrect statement out of these

I got this question from somewhere. It is easy but I am not able to find the incorrect option out of these. Let $A$ be a square matrix of order $n$. Let $m_A$ denote the minimal polynomial and $...
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Linear transformation and minimal polynomial

A linear transformation $T: \Bbb Z_p \times \Bbb Z_p \to \Bbb Z_p \times \Bbb Z_p $, where $p$ is a prime, is defined by $T(v)=s^{-1}vs$. The minimal polynomial is $m(x)=x^2+x+1$. Also $u=T(t)=s^{-1}...
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1answer
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Minimal Polynomial of null operator

I think that is a simple question, but there is some details confusing me. I need to calculate the minimal polynomial and the characteristic polynomial of the null operator in a $\mathbb{F}$-space. My ...
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1answer
29 views

Constant term in a minimal polynomial is a scalar, but not so when polynomial is composed with linear transformation.

So if the minimal polynomial of some linear transformation is say $\ \mu (x) = x^2+x+2$, then if we put in a matrix $\ A$ in for $\ X$ instead, we would write this is as $\ \mu(A) = A^2 + A + 2I$ with ...
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Prove that if every linear operator which commutes with $T$ is a polynomial in $T$, then $T$ admits a cyclic vector $v$ in $V$ (finite dimensional). [duplicate]

Let $T:V→V$ be a linear map over a finite dimensional vector space V with $dim$ $V=n$. Prove that if every linear operator which commutes with $T$ is a polynomial in $T$, then T has a cyclic vector $...
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Find the minimal polynomial over $\mathbb{Q}(i)$

Find the minimal polynomial $\exp\left(\frac{2\pi i}{5}\right)$ over $\mathbb{Q}(i)$ It is clear, that polynomial $\exp\left(\frac{2\pi i}{5}\right)$ over $\mathbb{Q}(i, \sqrt{5})$ is $f(x) = x - \...
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1answer
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Show that $F(T)$ is invertible if and only if it has no common factors with the minimal polynomial of T.

Suppose $F(x)$ is a polynomial over $\mathbb{C}$ and that $T$ is a function from a finite dimensional complex vector space to itself. Show that $F(T)$ is invertible if and only if it has no common ...
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Clarification Regarding to prove diagonalisable operator iff Minimal Polynomial Splits and

I wanted to prove that A is diagonalisable over F iff minimal Polynomial of A splits over F and Have square free terms. In that proof, I had following doubts If A is diagonalisable then it is clear ...
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1answer
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Find minimal polynomial of matrix with some given property

Let $A$ be a $4 \times 4$ non-diagonizable complex matrix that satisfies the relation $A(A-2I_4)^2=0$. I want to find, with proof, all the possible forms that the minimal polynomial $m_A(x)$ of the ...
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1answer
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Minimal polynomial for module over $A$ such that $A[x]$ is not a PID

Given an endomorphism $T$ of $V$ a vector space over a field $k$, we may define the minimal polynomial of $T$ to be a generator of the annihilator of $T$ in $k[x]$. We are guaranteed such a unique ...
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1answer
54 views

$A^3 = A^2$ How can $A$'s minimal polynomial look like?

Let $K$ be a field and $A \in K^{n \times n}$ a matrix with $A^3 = A^2$. How can $A$'s minimal polynomial $\mu_A$ look like? The only possibilities I could think of are $A = 0$. Then the ...
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Rewriting a sum of harmonic powers as a minimal polynomial

Revisiting one of my older questions, I've decided to try to tackle a simpler version of the problem, this time without the square root coefficients. Let $x_0$ be a real number such that it ...
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1answer
38 views

Prove or disprove that $[L:K] = \deg(p)$ [duplicate]

Let $L/K$ be a finite extension and let $\beta \in L$. If $p$ is the minimal polynomial of $\beta$ then is it true that $[L:K] = \deg(p)$? If not, give a counterexample. Can someone help me figure ...
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Can the minimal polynomial of a matrix have a root with multiplicity?

Can the minimal polynomial $P$ of a square matrix over some field $F$ have the form $P=Q(X-\lambda)^2$ for some $\lambda \in F$ ?
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1answer
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Difficulty finding minimal polynomial…

I would like to find the minimal polynomial of the following matrix. $$ A = \begin{pmatrix} 1& 2& 0& 0 \\ 2& 1& 1& 1 \\ 0& 0& 1& 2 \\ 0& 0& 0& 1 \end{...
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Showing the minimal polynomial of $\sqrt[n]{p}$ over $\mathbb{Q}$ is $x^n-p$

Let $p$ be a prime and $n$ a positive integer. I am trying to understand why the minimal polynomial of $\sqrt[n]{p}$ over the rationals must be $x^n-p$. I can show this in the case where $n=2$ but am ...
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Given square real matrix $A$ with $\det(A) = 108$ and $(A-2I)(A^2-9I)=0$, is $A$ normal?

Given real square matrix $A$ with $\det (A) = 108$ and $(A-2I)(A^2-9I) = 0$, find: a. The characteristic and minimal polynomial options (all options). b. Is $A$ normal? I think I found the ...
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Constant Term of a Monic Polynomial in $\mathbb{Z}[x]$ Is Divisible by 3

Given a monic polynomial $f(x)$ in $\mathbb{Z}[x]$ such that $\alpha$ and $3 \alpha$ are complex roots of $f(x),$ prove that the constant term of $f(x)$ is divisible by 3. I have attempted this ...
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1answer
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are parabolic points in the Mandelbrot set algebraic numbers?

Define the iterated quadratic polynomial: $$ \begin{aligned} f_c^0(z) &= 0 \\ f_c^{n+1}(z) &= f_c^n(z)^2+c \end{aligned} $$ The $c$ for which $f_c^n(0)$ remains bounded form the Mandelbrot ...
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Seeking the minimal polynomial for an algebraic monomial

I have an algebraic monomial relationship: \begin{equation}\tag{1} A_1c^{12} + A_2c^{10} + A_3c^8 + A_4c^6 + A_5c^4 + A_6c^2 + A_7 = 0 \end{equation} where $A_1 - A_7$ are algebraic numbers ...
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How to show linear independence of three elements connected by a linear transformation.

Let $V$ be a three dimensional vector space over the field $\mathbb{Q}.$ Suppose $T: V \to V$ be a linear transformation and $T(x)=y,$ $T(y)=z,$ $T(z)=x+y,$ for certain $x,y,z \in V,x \neq0.$ Prove ...
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1answer
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Degree of a field extension of a minimal polynomial

Suppose a polynomial $f(x)$ of degree $n$ over $\mathbb{Q}$ is the minimal polynomial of an element $\alpha$ in an extension field of $\mathbb{Q}$. Is $[\mathbb{Q}(\alpha):\mathbb{Q}]=n$? Please give ...
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1answer
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Showing the minimal polynomial for an element in an extension field is the same as the minimal polynomial of a linear transformation.

This is problem 31b) from Dummit and Foote chapter 14.2. I am looking for a hint on how to attack the problem, since I have been thinking about it for a couple of hours but I don't even know where to ...
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Conditions for Which $\mathbb{F}[T]$ Is a Field

Given a linear operator $T$ on a finite-dimensional vector space $V$ over a field $\mathbb{F},$ consider the ring $\mathbb{F}[T]$ of polynomials in the linear operator $T.$ Under what conditions is $\...
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Minimal polynomial generators of aurifeuillan factors cyclotomic polynomials

In order to answer this question, I came up with a way to generate the reciprocals of aurifeuillan factors of $n$-th cyclotomic polynomials for odd prime $n$. If $n=1\pmod 4$, then $\Phi_n(nx^2)$ has ...
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If $f^2=f$, then $f$ is of constant rank.

Let $f$ be a continuous function from $[0,1]$ to set of $n\times n$ matrices i.e. $M(n\times n,\mathbb{R})$ such that $f(t)^2=f(t)$ for all $t$. Then $f(t)$ has a constant rank for all $t$. The only ...
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Degree of Field Extension $[\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}]$

My problem is understanding how we relate field extensions with the same minimum polynomial. I am running into some problems understanding some of the details of the field extension $\mathbb{Q}(2^{\...
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Does the Minimum Polynomial of a Matrix depend continuously on the entries of the matrix?

We know that the coefficients of the characteristic polynomial of a matrix depend continuously on the entires of the matrix (since they are polynomials in the entires of a matrix). The coefficients ...