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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What is the minimum degree of polynomial with $\sqrt[n]{a}$ as a root that has integer coefficients?

I am looking for the minimum degree polynomial with integer coefficients that has $\sqrt[n]{a}$ as a root where $n$ and $a$ are integers, and $a$ is not a $n$-th root, i.e. $\sqrt[n]{a}$ is irrational....
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Degree of dual curve of $x^d+y^d+z^d=0$

My attempt: The dual curve of $x^d+y^d+z^d=0$ is the minimal polynomial of $x^{d/(d-1)}+y^{d/(d-1)}+z^{d/(d-1)}$ $d$ minimal polynomial degree of polynomial $2$ $x^2+y^2+z^2$ $2$ $3$ $x^6+ y^6+ z^6 ...
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Proving $\text{Aut}(\mathbb{Q}(i, \sqrt{2})) \cong V_{4}$ and finding minimal polynomial

Let $M = \text{Aut}(\mathbb{Q}(i, \sqrt{2})) \cong V_{4}$ and show that the $\prod_{\sigma \in M}(X - \sigma(\alpha))$ is equal to the minimal polynomial of $\alpha = 1 + i + \sqrt{2}$ over $\mathbb{Q}...
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Show degree of minimal polynomial has to be even

We define the polynomial $f(T) := T^6 - 3T^2 + 1 \in \mathbb Q[T].$ Let $\alpha \in \mathbb C$ be a root of $f.$ I showed that $f \mod 3$ can be factorized into three irreducible factors of degree $2:$...
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Is kernel of evaluation map on $R[x]$ always principal if $R$ is an integral domain?

Suppose $R$ is an integral domain and $E \supseteq R$ is a ring extension of $R$. Let $\alpha \in E$ commute with all of $R$, and consider the evaluation homomorphism $\phi_\alpha : R[x] \to E$ given ...
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Prove that $V_{\lambda_1}\subseteq \operatorname{ker}\rho$.

I'm trying to prove the following theorem: Let $V$ be a finite-dimensional vector space over an algebraically complete field and let $T$ be a linear operator on $V$ with minimal polynomial $$m_T(x) = \...
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A $4\times 4$ matrix counterexample. [duplicate]

A question in Dummit & Foote is asking to prove that two $3\times 3$ matrices are similar iff they have the same characteristic and the same minimal polynomial. I was able to prove that. But then ...
Hope's user avatar
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Direct Sum of Factors of Minimal Polynomial

Let $T$ be a linear operator on a finite-dimensional vector space $V$ over $\mathbb{C}$. Assume that the minimal polynomial $m(z)$ of $T$ factors as $m(z) = a(z)b(z)$, where $a(z)$ and $b(z)$ are ...
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Why does the minimal polynomial of $T\restriction_{\operatorname{ker}(\rho)}$ divide $a$?

Let $T$ be a linear operator on a finite dimensional vector space $V$. If we write its minimal as a product of coprime monic polynomials $$m_T(x) = a(x)b(x)$$ then we can write $$1 = \alpha(x)a(x) + \...
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Confusion about the primary decomposition theorem in linear algebra

I am currently studying the Jordan canonical form which uses the primary decomposition. I have seen the generalised eigenspace decomposition and I know that the algebraic multiplicity which appears in ...
Wintermelon423's user avatar
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given the minimal polynomial of matrix $A^2$ find minimal polynomials and Jordan forms of $A$

I'm refreshing some linear algebra and came across this question: Consider a matrix $A \in \mathbb{C}^{4\times 4}$ and suppose the minimal polynomial of its square is $\phi_{A^2}=(x-1)^2$. Now we're ...
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Classifying Matrices

Determine up to similarity all $3 \times 3$ complex matrices $A$ such that $A^4 + 2A^3 + A^2 = 0$ and $A^2 + A \neq 0$. Give the characteristic and minimal polynomial of each matrix. I'm not quite ...
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Question about matrix with irreducible minimal polynomial

Let $M$ be a matrix over $\mathbb Q$ with an irreducible minimal polynomial $m_M$. Suppose we have a polynomial $P \in \mathbb Z[x]$ such $P(M)(e_i) = \underline{0}$ for some $i$, i.e. the matrix $P(...
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Find the minimal polynomial for a root over the rationals

Find the minimal polynomial over $Q$ of the root $(\sqrt[2]{3})/(1+\sqrt[3]{2})$. I'm new to field theory so please help! (I'm new to the platform so excuse my sloppy notation) So far, I set a= $(\...
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Confusing Argument about Minimal and Characteristic Polynomials in Hoffman-Kunze's Linear Algebra

I was confused about the following argument; not listed as a proof of anything but in one of the explanatory blurbs, on Page 193 of Hoffman-Kunze, in $\S 6.3$. Let $T$ be a diagonalizable linear ...
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When do $\cos \theta$ and $\sin \theta$ exist over $\mathbb{F}_p$

Let $p$ be a prime and $\theta \in \left[0, \frac{\pi}{2}\right]$ be a real number. Suppose $\cos \theta$ and $\sin \theta$ are algebraic over $\mathbb{Q}$. When do they also exist over $\mathbb{F}_p$,...
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Need help solving this minimization function of jerk by use of Euler-Poisson equation

I need help solving this. In the picture below a paper explains why that the jerk (third derivative of position) is already cost optimal. So this means that a sixth order polynomial is already jerk ...
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Computing the Galois group of polynomials of high degree

I have the following polynomial $f(x)=x^6-15x^4-14x^3 + 75x^2-210x-76$ for which I know that $\sqrt[3]{7} + \sqrt{5}$ is a root. I guessed $\sqrt[3]{7} - \sqrt{5}$ is also a root, and most probably ...
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Finding the minimal monic polynomial given a root [closed]

Given $z = \sqrt[3]{7} + \sqrt{5} \in \mathbb{R}$ a root, compute its minimal monic polynomial $f(x)\in \mathbb{Q}[x]$. Is there a clever way to do this without having to compute $z^2$, $z^3$, $z^4$, $...
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How to find minimal polynomial if we know some root [duplicate]

So I tried to ask this question about two days ago, but no one could help me and I was wrong with some terminology, and my question was deleted. Im studied abstract algebra and trying to solve some ...
Danila Rudnev's user avatar
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Question about the degree of the splitting field of $X^4-2X^2-2$ [duplicate]

I have one question regarding an exercise that I have done: The exercise is the following: For the following polynomial from $\mathbb{Q}[X]$, determine a splitting field in $\mathbb{C}$ and the degree ...
Marco Di Giacomo's user avatar
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Factor $x^5+x^4+x^3+x^2+x+1$ over $\Bbb F_5$

This is the polynomial with roots $\omega,\omega^2,...,\omega^5$, where $\omega$ is the primitive $6^{th}$ root of unity over $\Bbb F_5$. So that the irreducible factors are the minimal polynomials of ...
user108580's user avatar
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Discriminant divisible by prime iff minimal polynomials reduction mod p has multiple roots

I am attending a first course in algebraic number theory. We have learned the basics of field extensions, integral closures, norm and trace. I am trying to solve the following problem: Let $K=\mathbb{...
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Minimal polynomial requirements

I'm a bit confused with how we can show a given polynomial that annihilates a linear transformation $T$ is a minimal polynomial. In one particular problem, I have the polynomial $x^m - 1$ annihilates ...
Guilherme Souza's user avatar
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How to use the minimal polynomial theorem for this block matrix

Let $M$ be a matrix made up of two diagonal blocks: $M = \begin{pmatrix} A & 0 \\ 0 & D\\ \end{pmatrix}$ Prove that $M$ is diagonalizable if and only if A and D are diagonalizable. I know I'd ...
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Relation between the degree of minimal polynomial and rank of a matrix

I have seen the following True/ False question If $A ∈ M_n(\mathbb{R})$ (with $n \geq 2$) has rank 1, then the minimal polynomial of $A$ has degree 2. The statement is true. If we generalize the ...
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Minimal Polynomial Degree Inequality for Linear Operator and Its Square

I found a problem I got stuck with: Let $f \in \operatorname{End}(V), V$ be a finite-dimensional $F$-vector space, $F$ algebraically closed. (a) Let $ a \in F $ and $k \in \mathbb{N}$. Show: If $(f-a)^...
Marius Lutter's user avatar
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Let A be a complex $n \times n$ matrix with rank(A) = 1. Why is the minimal polynomial $x(x-Tr(A))$? [duplicate]

We know $rank(A) = 1$ so I have $n-1$ eigenvalues which are $0$ So my characteristic polynomial is $(x-0)^{n-1}(x-a) = x^{n-1}(x-a)$ with $a$ the last eigenvalue to determine. Now, I found on some ...
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Minimal polynomial of $\operatorname{cd}\frac{4K}{n}$

Define $$K=\int_0^1 \frac{dx}{\sqrt{1-x^4}}$$ and the Jacobi elliptic function $\operatorname{cd}$ with modulus $i$ by $$\int_{\operatorname{cd}z}^1\dfrac{dx}{\sqrt{1-x^4}}=z$$ on $z\in [0,2K]$ and by ...
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How are eigenspaces of an vector space endomorphism related to the decomposition as a direct sum of abelian groups? [duplicate]

The context of my question is the following problem Let $V=\mathbb Q^7$, $\phi: V \rightarrow V$ the $\mathbb Q$-linear map given by the matrix: $A=\begin{align*} \begin{pmatrix} 1&0&0&0&...
some_math_guy's user avatar
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Theoretical justification of method to find the minimal polynomial of $\alpha=\sqrt p + \sqrt q$ over $\mathbb Q$ with $p,q $ distinct prime numbers

Find the minimal polynomial of $\alpha=\sqrt p + \sqrt q$ over $\mathbb Q$ with $p,q $ distinct prime numbers My approach when I saw this problem was to just use some high-school algebra. I recalled a ...
some_math_guy's user avatar
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minimal polynomial of a complex number in Mathematica [closed]

How do I calculate the minimal polynomial of $a+ \mathcal{i} b$ where $a,b \in \mathcal{R}$ in Mathematica. In Mathematica, if I give specific values of a and b, then it gives the solution, for ...
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Determine the degree of a root of $X^p-X-\alpha$ (Serge Lang Algebra exercise VI.29)

Let $K$ be a cyclic extension of a field $F$, with Galois group $G=\langle \sigma \rangle$ and assume that $\operatorname{char}F=p$ and that $[K:F]=p^{m-1}$ for some $m>1$. Let $\beta$ be an ...
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How do we find the conjugates of a field element over $GF[2^{4}]$? And why can we take the LCM of just odd-indexed min.polynomials for g(x)?

I am following an example given by Writi M on a youtube video "an example of construction of BCH codes and encoding using BCH codes" The example asks us to construct a BCH code that can ...
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Common eigenvector for 3x3 complex matrix with minimal polynomial of degree at most 2 [duplicate]

Let $T_1, T_2: \mathbb{C}^3 \rightarrow \mathbb{C}^3$ linear maps. Show that if both linear maps have minimal polynomials of degree at most 2, then there is a vector that is an eigenvector for both $...
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Degree and Galois group of $\mathbb{Q}(\sqrt[3]{2},\sqrt{-3})/\mathbb{Q}(\sqrt{-3})$.

I would like to find the degree and Galois group of $\mathbb{Q}(\sqrt[3]{2},\sqrt{-3})/\mathbb{Q}(\sqrt{-3})$. I usually can compute the degree of an extension over $\mathbb{Q}$ as one only needs to ...
Yang Awotwi's user avatar
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Characteristic polynomial and determinant of block matrices

If the square matrix $A$ is invertible, $$\det\begin{bmatrix}A&B \\ C&D\end{bmatrix} = \det(A) \det\left(D-CA^{-1}B\right)$$ Using this formula, can we come up with the characteristic ...
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Why are the kernel of the irreducible factors of the minimal and charateristic polynomial equal?

If $\varphi$ is an Endomorphism over $V$ with $\mu_\varphi=\prod^n_{i=1}p_i^{m_i}$ for some $n,m_i$ and irreducible $p_i$ and $\chi_\varphi=\prod^n_{i=1} p_i^{c_i}$ with $c_i\geq m_i$ then $\...
watertrainer's user avatar
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Classification of all similar $3\times 3$ matrices over $\mathbb{R}$

I want to ask a question regarding classes of similar $3 \times 3$ matrices. We were told that two matrices are similar if and only if they have the same Jordan normal form. That led me to trying to ...
watertrainer's user avatar
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number of extensions of a field embedding after taking a finite extension of the domain

I'm learning Galois theory, and trying to prove the following theorem: Let $K$ be a field and $L/K$ be a finite extension with $x_1,\dots,x_n\in L$ such that $L=K(x_1,\dots,x_n)$. let $M$ be a field ...
jeff honky's user avatar
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A question on matrices whose minimal polynomials has no repeated roots

Question a) Prove that a matrix $A\in M_n(\mathbb{C})$ satisfying $A^3=A$ can be diagonalized. b) Does the statment in (a) remain true if one replaces $\mathbb{C}$ by an arbitrary algebraically closed ...
confused's user avatar
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prove $\deg_{\alpha,L}\mid\deg_{\alpha,K}$ if $\alpha$ is separable and $K(\alpha)/K$ is normal

I'm trying to prove the following: Let $K$ be a field, $\alpha \in \overline{K}$ a separable element s.t. $K(\alpha)/K$ is normal, and let $L/K$ be some finite subextension of $\overline{K}$. Prove ...
Ariel Yael's user avatar
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Minimal polynomial of block triangular matrix

If $$T=\begin{pmatrix}A&0\\0&C\end{pmatrix}$$ is a block diagonal matrix, then (I assume) we have \begin{equation}\tag {1}m_T=\text{lcm}(m_A,m_C)\end{equation} for the minimal polynomials. If ...
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Question about calculation of the minimal polynomial [duplicate]

Let $\alpha=\sqrt3+\sqrt5$, I want to calculate the minimal polynomial of $\alpha$ over $\mathbb{Q}$. The Idea seems to be to determine the smallest $n \in \mathbb{N}$ such that $\{1,\alpha,...,\alpha^...
Sigi's user avatar
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Prove that if $\phi: V \to V$ is diagonalizable and $U \subseteq V$ is a $\phi$-stable subspace then $\phi|_U$ is also diagonalizable. [duplicate]

So we have a field K and a finite K-Vectorspace V. $\phi: V \to V$ is a linear function. How can I prove the question in the title? We've been told to use the following theorem: A matrix A is ...
Toilet Paper's user avatar
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Find the dimension of $\mathbb{Q(\sqrt{5},i)}$ over $\mathbb{Q}$ [duplicate]

What is the dimension of $\mathbb{Q(\sqrt{5},i)}$ over $\mathbb{Q}?$ Considering that $\mathbb{Q} \subseteq \mathbb{Q(\sqrt{5})} \subseteq \mathbb{Q(\sqrt{5},i)}$ $[\mathbb{Q(\sqrt{5},i)} : \mathbb{Q}]...
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Dimension of $\mathbb{Q(\omega)}$ and minimal polynomial of $\sqrt[3]{2}$

Consider: $$\omega = \frac{-1}{2} + \frac{\sqrt{3} i}{2}$$ and the simple extension $\mathbb{Q(\omega)}$. Find the dimension of $\mathbb{Q(\omega)}$ and the minimal polynomial of $\sqrt[3]{2}$ over $\...
jontao's user avatar
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2 votes
1 answer
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Different algebraic expressions for the same roots of a polynomial. [duplicate]

The polynomial $X^4-10X^2+1$ has roots $\pm\sqrt 2\pm\sqrt 3$. My CAS, however, returns the roots as $\pm\sqrt{5\pm2\sqrt 6}$. Probably because it uses the quadratic formula (after substituting $X^2$)....
YellowCake's user avatar
2 votes
1 answer
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Cube roots of the identity matrix

Describe the set of cube roots of the identity matrix in $M_{n \times n}(\mathbb{R})$. My attempt: Let $I_n$ denote the identity matrix of dimension $n \times n$. To find the cube roots of $I_n$, one ...
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Minimal Polynomial of $\sqrt[3]{2}$ over $\mathbb{Q}(e^{\frac{2\pi i}{3}})$ and degree of field extension.

Hello and thanks in advance for any responses. I'm stuck on the following problem: "Show that the polynomial $X^3 - 2$ is irreducible over $\mathbb{Q}(\omega)$, where $\omega = e^{\frac{2\pi i}{3}...
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