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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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If $\mu_T = pq$ non-trivially, is $\mathrm{Ker}\: q(T)$ the only $T$-invariant complement to $\mathrm{Ker}\:p(T)$?

Context Some time ago, I found out that the minimal polynomial gives us a nice way to decompose a finite dimensional vector space $V$ into a direct sum of invariant subspaces: Lemma Let $V$ finite ...
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15 views

Let $P$ be a $n\times n$ matrix, if there is $k\in \mathbb Z^+$ such that $P^k=O$, prove that $P^n=O$. [duplicate]

Let $P$ be a $n\times n$ matrix, if there is $k\in \mathbb Z^+$ such that $P^k=O$, prove that $P^n=O$. I have thought about characteristic polynomial, but it doesn't give me much information. So I ...
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Simplicity of the roots of a minimal polynomial

Let $L/K$ be a finite field extension, and let $\mu_{\alpha,K}\in K[X]$ be the minimal polynomial of $\alpha\in L$. One can easily see that $\alpha$ is a simple root of $\mu_{\alpha,K}$. Indeed, if ...
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1answer
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Computing minimal polynomial in finite field F8

In a finite field $F_q$, I've read that one can get the minimum polynomial $f(z)$ of an element $\beta \in F_q$ using this formula: $$f(z) = (z-\beta)(z-\beta^2)(z-\beta^4)(z-\beta^8)...$$ I'm ...
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1answer
27 views

A question regarding finding the minimal polynomial associated with a field extension . [duplicate]

Say we have the field extension $\Bbb Q(w,\sqrt[3]{5})$ over $\Bbb Q$, where w is the primitive cubed root of unity. I know that the minimum polynomial of $\sqrt[3]{5}$ is $x^3-5$. I want to figure ...
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1answer
24 views

Which polynomial kernels are cyclic, and how to find a cyclic generator

Let $T$ be a $\Bbbk$-linear operator on a vector space $V$. Given $f\in \Bbbk[x]$ consider the operator $f(T)$ on $V$. Write $K(f)=\operatorname{Ker}f(T)$ and call such $T$-invariant subspaces ...
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3answers
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On Minimal polynomial and Invariant subspace

Given $T:V \rightarrow V$, $V$ is a vector space over field $\mathbb{R}$ and $m_T = (x^2-2x+2)(x-3)^2$. Show that there exists an invariant subspace with dimension $2$. I first thought that since $3$ ...
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2answers
201 views

Why is $(X^5-1)/(X-1)$ the minimal polynomial for $e^{2\pi i/5}$, the fifth root of unity, over $\Bbb Q$?

I have some trouble understanding an example from my reader$^1$. We have the fifth root of unity $\zeta_5=e^{2\pi i/5}$, so by definition $\zeta_5^5=1$. Then I want to find its minimal polynomial ...
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1answer
28 views

characteristic polynomial with rows as coordinates

Suppose $V$ is a finite dimensional vector space with basis $E$, $f$ an endomorphism of $V$ and $B$ another basis of $V$. Denote by $B'$ the matrix whose rows are the coordinates of $B$ and by $C'$ ...
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Find the characteristic and minimal polynom of the following transformation

The transformation is $$T:M_{\leq n}(\mathbb{C})\rightarrow M_{\leq n}(\mathbb{C})$$ and defined by $$T(A) = A^t-A.$$ If we try to take a basis $B$ and calculate the determinant of $xI-[T]_B$, we get ...
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1answer
17 views

Finding minimal polynomial with given operator

Given the operator $T:\mathbb C_{\le n}[x]→\mathbb C_{\le n}[x]$ such that $T(p) = p' + p$ find the minimal polynomial. What I tried: I found the representing matrix $$A = \begin{pmatrix} 1 & ...
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1answer
41 views

How to prove the uniqueness of the minimal polynomial?

I have here the definition: Let $T$ be a linear operator on a finite-dimensional vector space $V$ over the field $F$. The minimal polynomial for T is the (unique) monic generator of the ideal ...
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1answer
23 views

Does this method always produce the minimal polynomial over $\Bbb{Q}$?

I have used it a few times and so far I have not seen an instance in which it did not give me the minimal polynomial over $\Bbb{Q}$. The method is the following: For an element $a \notin \Bbb{Q}$ set ...
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1answer
51 views

Show that matrix $A$ is similar to a matrix $B$ with elements on diagonal $(0, …, 0, \operatorname{Tr(}A))$ respectively.

Let $A$ be a matrix $n \times n, n \geq 2 $. Let's assume that not all entries outside of the diagonal are zeros (we don't know what entries are on the diagonal). Show that matrix $A$ is similar to a ...
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1answer
48 views

Determine the minimal polynomial of $\sqrt{2+\sqrt{2}}$ over $\Bbb Q$ and find its Galois group over $\Bbb Q$

Determine the minimal polynomial of $\sqrt{2+\sqrt{2}}$ over $\Bbb Q$ and find its Galois group over $\Bbb Q$. Computed and obtained $p(x)=x^4-4x^2+2$ has $\sqrt{2+\sqrt{2}}$ as a root. $p$ is ...
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1answer
51 views

Irreducible characteristic polynomial and invariant subspaces

Let $K$ be a field and let $V$ be a finite-dimensional vector space over $K$. Let $\alpha$ be an endomorphism of $V$, with irreducible characteristic polynomial. I'm trying to show that there is no ...
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1answer
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A question about Integral Closures from Hartshorne Chapter 2 Exercise 6.4

The exercise that I'm having trouble with is the following. Hartshorne II.6.4: Let $k$ be a field of characteristic $\neq 2$. Let $f \in k[x_1, \dots x_n]$ be a square free nonconstant polynomial, i....
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0answers
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If $f_{K(\alpha)}^{\beta}\big| f_{K}^{\beta}$, then $\deg\left(f_{K(\alpha)}^{\beta}\right)| \deg\left(f_{K}^{\beta}\right)$?

Let $K\subset K(\alpha)\subset K(\alpha,\beta)$ be field extensions. Then the question is whether the minimal polynomials of $\beta$ over $K$ and $\beta$ over $K(\alpha)$ have degrees where one ...
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41 views

Finding a minimal polynomial satisfying a few conditions

I have solve an exercise that ask: Find the minimum degree polynomial $P(x) \in \Bbb R[x] : P(1+2i) = 0 \land P(i) = i$. The solution is quite obviously: $P(x) = \frac{1}{10}(x^2 -2x +5)(2i-1)$. ...
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1answer
31 views

Verifying a $K$-basis for a primitive extension $K\subset K(\alpha)$

Let $K\subset L:=K(\alpha)$ be a primitive field extension of degree $n$ and we define $c_i\in L$ as \begin{align*} \sum_{i=0}^{n-1}c_i x^i=\frac{f^{\alpha}_K}{(x-\alpha)}\in L[x]\quad(1) \end{...
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1answer
30 views

Questions about proof of existence of roots of $f$ in $K[X]/(f)$

Let $f \in K[X]$ with $deg(f)\geq 1$. Then there exists an algebraic field extension $L/K$, such that $f$ has a root in $L$. Proof: WLOG we can assume that $f \in K[X]$ is irreducible. Since $...
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2answers
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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
17 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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0answers
45 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
23 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
43 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
30 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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1answer
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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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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
33 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
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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
53 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
92 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
35 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
35 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
26 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
50 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
79 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
59 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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55 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
57 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
41 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
42 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
94 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-...
3
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1answer
100 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
57 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
93 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
23 views

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
190 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 ...