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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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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 detrminant

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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21 views

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
26 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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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
34 views

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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31 views

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
48 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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1answer
51 views

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
33 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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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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1answer
38 views

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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62 views

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
59 views

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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1answer
56 views

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 ...
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Jordan normal form options from minimal polynomial

what is the Jordan normal form options from this minimal polynomial 𝑚𝐴(𝑥) = $𝑥^3 − 2x^2$. 4X4 matrix I know of course that the 2 eigenvalues are 0,2.
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Minimal polynom for $e^{2\pi i/3} + 2^{1/3}$

I have $\alpha = \exp(\frac{2\pi i}{3}) + 2^{\frac13}$ and I need to find a minimal polynomial for It in $\mathbb Q$, the "obvious" Path is to get the third Power of each term, but I'm struggling with ...
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1answer
31 views

Can one describe the algebraic multiplicity in terms of generalized eigenspaces and the minimal polynomial?

We defined the algebraic multiplicity of a matrix $A$ with eigenvalue $\lambda$ to be the largest integer $r$ such that $(x-\lambda)^r$ divides the characteristic polynomial of $A$. I would like to ...
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2answers
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Have I found the Jordan form correctly?

I am given that the minimal polynomial and characteristic polynomial of a matrix are both $(x-1)^2(x+1)^2$. I have found the Jordan form to be $$\begin{bmatrix}1&1&0&0\\0&1&0&0\...
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3answers
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How To Find Minimal Polynomial

$$A=\left(\begin{array}{ccccc} 4 & 1 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 & 0 \\ 0 & 0 & 4 & 0 & 0 \\ 0 & 0 & 0 & 9 & 0 \\ 0 & 0 & 0 &...
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Are there uncountably many $A$ such that $A^8=I $?

I'm working on the following problem: Let $A \in M_3 (\mathbb {R})$ be such that $A^8=I$. Then the minimal polynomial of $A$ can only be of degree $2$. the minimal polynomial of $A$ ...
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Compute $T_{E|\mathbb{Q}}(\alpha)$ and $N_{E|\mathbb{Q}}(\alpha)$ for every $\alpha \in E$.

Consider the splitting field $E$ of $X^4 -2$. Compute $T_{E|\mathbb{Q}}(\alpha)$ and $N_{E|\mathbb{Q}}(\alpha)$ for every $\alpha \in E$. My attempt: The splitting field $E = \mathbb{Q}(i,\sqrt{2},\...
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1answer
54 views

Find minimal polynomial of element

Find minimal polynomial of element $3-2\sqrt[3]{2}-\sqrt[3]{4}$ $\in {\displaystyle \mathbb {Q}(\sqrt[3]{2}) } $ over the field $ {\displaystyle \mathbb {Q} } $. Thank you for any help.
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Prove that this linear transformation is nilpotent

The Question: Let $V$ be a finite dimensional vector space over $\Bbb C$, with linear map $T:V \rightarrow V$. Suppose that the minimal polynomial of $T$ is $$m_T(x) = p(x)q(x) \\ p(x)=(x-\lambda)^\...
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1answer
75 views

Find smallest $A\in \mathbb{R}$ such that $|f'(0)|\leq A.$

Problem Find smallest $A\in \mathbb{R}$ such that for 3-degree polynomial satisfying the condition $\forall x\in[0;1],\ |f(x)|\le 1$ holds $|f'(0)|\le A$. Solution Assume that the 3-degree ...
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1answer
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Where is the flaw in my reasoning about the number of irreducible degree 9 polynomials over $\mathbb F_2$?

I am aware of the formula for the number of polynomials of degree $n$ over a finite field, and that it gives 56 for this particular case, but I wanted to know why the following train of thought is ...
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What is the Galois group of $f = X^4 - 3X^2+3 \in \mathbb{Q}[X]$?

The polynomial $f$ is an irreducible Eisenstein polynomial with $p = 3$. Its roots are easy to find using the substitution $Y = X^2$ and then the $abc$-formula: $\{\sqrt{\frac{3}{2}+\frac{\sqrt{-3}}{...
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Does this fact about the minimal polynomial give an efficient diagonalizability criterion?

In a very nice paper "When Is a Linear Operator Diagonalizable?" by Marco Abate (Amer. Math. Monthly 104 (1997), 824-830) I found the following nice description of the minimal polynomial $\mu(T)$ of ...
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How to show that any matrix A over $M_{2x2}^{\mathbb{F}}$ is similar to $A^t$ without using Jordan blocks?

Is this true for any 2x2 matrix over any field? My approach that didn't led me to a solution. if $\begin{pmatrix}a&b\\ c&d\end{pmatrix}$ then we need a regular matrix $P$ such that $P^{-1}\...
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2answers
99 views

Jordan form of $15 \times 15$ matrix

Let $A \in M(15,15,\mathbb{R})$ be a matrix that satisfies: The characteristic polynomial is $p(x)=-x^5(x-1)^5(x+1)^5$ The dimension of the range of $A$ is $13$ and $\dim \ker A^2=4$. The dimension ...
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2answers
104 views

Minimal polynomial of $\sqrt{\sqrt[3]{7}-5}$

I'm trying to find the minimal polynomial of $\alpha = \sqrt{\sqrt[3]{7}-5}$. Rather, I know that it will be $p(x) = x^6 +15x^4+75x^2+118$ by just squaring and then cubing appropriately, but I would ...
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1answer
42 views

Check irreducibility of the polynomial

Problem: It is known that the minimal polynomial of an element $a \in F_4$ equals $x^2 + x + 1$. Does it follow from this that the polynomial $x^3 + ax^2 + a$ is irreducible in the ring of polynomials ...
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0answers
39 views

Minimal polynomial (lcm)/block diagonal matrix

Let $A=$ diag$(F_1,...,F_k)$ be a square matrix in block diagonal form. How to proof that the minimal polynomial of $A$ is $M_A=$lcm$(M_{F_1},...,M_{F_k})$? I tried with the proof that lcm$(M_{F_1},.....