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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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1answer
33 views

Is $A \in \mathcal{M}_n(\mathbb{C})$ diagonalizable?

$A \in \mathcal{M}_n(\mathbb{C})$ such that $A^2$ has got $n$ distinct non zero eigenvalues. Show that A is diagonalizable. Attempt : As $A^2$ has got $n$ distinct non zero eigenvalues. The ...
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1answer
13 views

Deducing if $\gcd(\deg(m_K(x)),\deg(m_K(y)))=1$ then $[K(x,y):K]=\deg(m_K(x))\times \deg(m_K(y))$.

I've shown that - If $x,y\in L$ are algebraic over $K$,then $[K(x,y):K]\le \deg(m_K(x))\times \deg(m_K(y))$. How can we deduce from the above result that if $$\gcd(\deg(m_K(x)), \deg(m_K(y)))=1,$$...
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31 views

field extension being algebraic is equivalent to every $K-$ algebra being an automorphism.

For a field extension $L\vert K,$show that the following statements are equivalent : $(i)$$L\vert K$ is algebraic. $(ii)$ For every $E\in${$E:E$ is a field with $K\subset E\subset L$},...
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1answer
20 views

Characteristic and minimal polynomial of A²

Let A be a $3\times 3$ integer matrix. If the characteristic polynomial of $A$ is equal to minimal polynomial of $A,$ does it follow that the characteristic polynomial of $A^2$ is equal to minimal ...
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1answer
38 views

Similarity of $2 \times 2$ matrices

If i am not mistaken, two $3 \times 3$ matrices are similar $ \iff \ $ they have the same characteristic and minimal polynomial. Also, if two $3 \times 3$ matrices have the same characteristic ...
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1answer
28 views

Determine field degree extension of certain number fields.

Let $p(x)\in \mathbb{Q}(x)$ be an irreducible monic polynomial of degree 3 and suppose all of its roots are real; call the roots $r_1,r_2,r_3$. Let $K=\mathbb{Q}(r_1,r_2,r_3)$. Is it the case that $K=\...
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A problem on algebraic extension in Galois theory

For a field extension $L\mid K$, I want to show that the following are equivalent: $L|K$ is algebraic. For every $E\in \mathfrak{F}(L|K)$, every $K$-algebra homomorphism $\sigma:E\to E$ is an ...
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8 views

Proving that the degree of transcendental extension is infinite

For a transcendental extension $K(\alpha):K$ for sub-field $K$ of $\mathbb{C}$, $[K(\alpha):K]=\infty$ Showing that the basis for $K(\alpha)$ that describes it as a vector space is infinite leads us ...
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1answer
30 views

$A$ and $B$ be $n \times n$ matrices over the field $\mathbb F$ which have the same characteristic polynomial

Lemma: Let $N_1$ and $N_2$ be $3 \times 3$ nilpotent matrices over field $\mathbb F$. Then, $N_1$ and $N_2$ are similar if and only if they have the same minimal polynomial. Use the result above ...
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3answers
28 views

Diagonalizable operator $D$ on $R^3$ and a nilpotent operator $N$

Let T be the linear operator on R3 which is represented by the matrix $\begin{pmatrix} 3 &1 &-1 \\ 2 &2& -1 \\ 2 &2& 0 \\ \end{pmatrix}$ in the standard ordered basis. ...
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2answers
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question about representation of the minimum polynomial

Let $T$ be a linear operator on $R^3$ which is represented in the standard ordered basis by the matrix $\begin{pmatrix} 6 &-3 &-2 \\ 4 &- 1& -2 \\ 10 &- 5& -3 \\ \end{...
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2answers
35 views

Isomorphisms of $\mathbb{R}[A]$ where $A$ is a $2\times2$ real matrix.

I'm trying to answer a question in which I'm supposed to show that if $A$ is a $2\times2$ real matrix then $\mathbb{R}[A]$ (the polynomials in $A$ with real coefficients) is isomorphic to one of: $$\...
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2answers
83 views

degree of a sum of two algebraic numbers

Let $a=\sqrt{2}$ and $b=\sqrt{3}$, then $a$ and $b$ are algebraic numbers of degree 2, while the degree of $a+b$ is not 2, actually it is 4, by the standard argument: if $x=a+b$, then we can ...
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1answer
34 views

Finding characteristic and minimal polynomials and the Jordan normal form of $f$, knowing some relations for $f$.

Given a vector space $V$ of dimension $4$ and a base $\{v_1,v_2,v_3,v_4\}$, let $f$ be an endomorphism of $V$ such that $f^3=0$ and moreover $f(v_1)=f(v_2)=v_3$, $f(v_3)=kv_4$, and $f(v_4)\in\left<...
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1answer
34 views

On $\bigg[\begin{matrix}E&O\\O&F\end{matrix}\bigg],B_b=\left[\begin{matrix}2b&-1&0&-1\\0&b&0&0\\0&-1&0&-1\\0&1&0&b\end{matrix}\right]\in M_4(\Bbb R)$

Consider the $4\times4$ matrices $A=\bigg[\begin{matrix} E&O_2\\O_2&F \end{matrix}\bigg]$, where $E,F$ are any nilpotent $2\times2$ matrices, and $B_b=\left[\begin{matrix}2b&-1&0&-...
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1answer
25 views

Minimal Polynomial under field Q

$ a={\sqrt 3 } - {\sqrt 5 } $ I have to find minimal polynomial for this under field Q after few squarings I got (1) $ f(x)= x^4-16x^2+4 $ And after I have factored out it to $ f(x)=(x - {\...
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1answer
48 views

Minimal polynomial?

Let $K$ a field, $p,n\in \mathbb{N}$, $B\in \mathcal{M}_p({K})$ and let denote $S_B=\{X\in \mathcal{M}_p(K) \ \mid \ X^n=B\}$. If $X\in S_B$, I have to prove that $\mu_X$ (the minimal polynomial of $...
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2answers
75 views

What is the minimal polynomial of $\sqrt{3+4i}+\sqrt{3-4i}$ over $\mathbb{Q}$?

My attempt: I know that $P=\mathbb{Q(\sqrt{3+4i}+\sqrt{3-4i})} \subseteq \mathbb{Q(\sqrt{3+4i},\sqrt{3-4i})}=K$. But I don't know whether $K \subseteq P$. Assuming $K=P$. I can see that K is Galois ...
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3answers
53 views

Minimal polynomial of a matrix having only 1s on the counter diagonal

Consider the matrix $A=a_{ij}$ where $$a_{ij}=\begin{cases}1\ \ \text{if}\ \ i+j=n+1\\0\ \ \text{otherwise}\end{cases}$$. Then, what can be said about the minimal polynomial of the matrix $A$. Note ...
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2answers
54 views

Is it 'rare' that $a$ and $a+1$ are conjugate (= have the same minimal polynomial)?

Let $a \in \bar{k}-k$, $k$ is a field of characteristic zero and $\bar{k}$ is an algebraic closure of $k$. Denote the minimal polynomial of $a$ by $m_a=m_a(t) \in k[t]$. Is it 'rare' that $m_a=m_{...
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0answers
55 views

$a,b \in \bar{k}$, such that $k(a)=k(b)=k(ab)$

Let $k$ be a field of characteristic zero, and let $a, b \in \bar{k}$ ($\bar{k}$ is an algebraic closure of $k$) be two distinct elements, such that $k(a)=k(b)$. Notice that $k(a)=k(b)$ implies that ...
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1answer
38 views

minimal polynomials of two elements $\alpha$ and $\beta$ that are algebraically independent.

Consider an extension of fields $K/F$ and $\alpha, \beta \in K$. If $\alpha$, $\beta$ are algebraically independent, is it true that the minimal polynomial of $\alpha$ (resp. $\beta$) on $K(\beta)$ (...
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0answers
37 views

minimal polynomial of a matrix B given minimal polynomial of $B^2$

If we are given a minimal polynomial for a matrix $B^2$ can we deduce the minimal polynomial for $B$ $?$ Example: if the minimal polynomial for $B^2$ is $m(\lambda) = \lambda^4$ then can we deduce ...
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2answers
86 views

Galois group of a certain splitting field

Let $f$ be the minimal polynomial for $\sqrt{3+\sqrt{2}}$. Find the Galois group of the splitting field $K$ over $\mathbb{Q}$. Here are the steps that I have taken. The minimum polynomial is $x^4-...
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1answer
97 views

Minimal polynomial algorithm

In our textbooks we are given the following algorithm: Let $V$ be a vector space having dimension $n$ over a field $K$ ( $\mathbb R$ or $\mathbb C$ ) and $A : V \to V$ be a linear map. In a sequence $...
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2answers
45 views

The minimal polynomial is the determinant of $xI-L_{\alpha}$.

Let $K=F(a)$ a finite field extension of $F$. For $\alpha \in K$, let $L_{\alpha} : K \to K$ be the transformation $L_{\alpha} (x)=\alpha x$. Show that $L_{\alpha} $ is an $F$-linear transformation ...
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1answer
86 views

If $A$ is an $n$ by $n$ integer matrix such that $A^3 = I$, then $\operatorname{tr}(A) = n\mod3$

Attempt: We work with $A'$, the matrix with entries $a_{ij}\mod 3$. Note that cubing $A'$ still gives $I$ as $I$ is unchanged by considering remainders $mod$ $3$. For the rest of the proof, we will ...
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1answer
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Clarification on proof about minimal polynomials

In Linear Algebra Done Right by Axler, Chapter 8C, there is a proof that states: Suppose $T \in \mathcal{L}(V)$ and $q \in \mathcal{P}(F)$. Then $q(T) = 0$ if and only if $q$ is a polynomial of $T$. ...
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4answers
186 views

Find all matrices which satisfy $M^2-3M+3I = 0$

I am trying to find all matrices which solve the matrix equation $$M^2 -3M +3I=0$$ Since this doesn't factor I tried expanding this in terms of the coordinates of the matrix. It also occurs to me ...
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0answers
39 views

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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24 views

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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0answers
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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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0answers
37 views

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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1answer
48 views

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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1answer
23 views

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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1answer
61 views

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 ...
2
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1answer
63 views

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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1answer
67 views

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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2answers
29 views

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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1answer
93 views

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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0answers
42 views

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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0answers
157 views

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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1answer
20 views

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 $\{...
5
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1answer
46 views

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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1answer
46 views

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
31 views

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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1answer
46 views

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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2answers
52 views

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 $\tau$ ...
2
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2answers
66 views

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 ...