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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.

For instance, $\sqrt2$ is an algebraic number, that is, it's a root of a non-zero polynomial with rational coefficients. Its minimal polynomial is $x^2-2$, since $\sqrt2$ is a root of this (monic) polynomial, and it is not a root of a non-zero polynomial with rational coefficients of a smaller degree.

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How to find the minimal polynomial of a matrix

Let $A=\begin{pmatrix} 1 &2 & 1 &0\\ -2 & 1 & 0 &1 \\ 0 &0 &1 &2 \\ 0 &0 & -2 & 1 \end{pmatrix}$. Then the characteristic polynomial of $A$ is $\chi (x)=(x^2-2x+5)^2$. I want to find the minimal polynomial of $A$. How can I find this? Are…
user112018
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How do I prove that this is the minimal polynomial of this particular root?

Let $x^n-b ∈ F[x]$ be an irreducible polynomial . Prove that $x^m−α^m∈F(α^m)[x]$ is the minimal polynomial of α over $F(α^m)$. It's so simple and apparently easy yet I can't say riguously why. I know that is equivalent to prove that $x^m-α^m$ is…
Jaqawa
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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 - {\sqrt 5 } + {\sqrt 3 } )( x - {\sqrt 5 } -{\sqrt 3…
Tovarisch
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The degree of a minimal polynomial

Is there in general any way of finding the degree of the factors in the minimal polynomial of an operator without using brute force computation? The exercise in question is shown below. It's only this part of (iii) that's bothering me. This is not…
Laplace's Demon
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Minimal Polynomial GF(5)

To calculate minimal polynomials in GF(2^m), I find conjugates and multiply. I am confused about minimal poly in a different GF. p(x)=x^3+x^2+1 over GF(5). It is irreducible over GF(5) and generates GF(5^3). I have proved that it is primitive (124…
user100503
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Problem about finding the minimal polynomials and its degree

I saw this exercise in an old exercise sheet and it's driving me nuts. There it is: For the first part, I think I have to apply the tower theorem and then I also know a property that says that the degree of the minimal polynomial of $a \in K$ over…
Jaqawa
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minimal polynomial of $\beta \not\in F(\alpha)$ is same as it in $F$?

Give a field $F$, let $\alpha$ is an algebraic element of $F$, let $f(x), g(x)$ are minimal polynomials of $\beta$ over $F$, $\beta$ over $F(\alpha)$, respectively. If $\beta \not\in F(\alpha)$, can I say $f(x)=g(x)$? I know it is true when…
xunitc
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minimal polynomial of a surd given the minimal polynomial of the radicand

Given an arbitrary radical expression $a$ whose minimal polynomial over $\mathbb{Q}$ is known to be $P(x)$, $P(x^n)$ is an annulling polynomial for $\sqrt[n]{a}$, where $n\in\mathbb{N}$. When $P(x^n)$ is not irreducible, is there an algebraic way to…
Alex Kindel
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finding minimal polynomial for $\frac{\theta^5}{5}$ when minimal polynomial for $\theta$ is $x^3-2$

suppose $f(x)= x^3-2$ is the minimal polynomial for $\theta$. How can I find a minimal polynomial for $a = \frac{\theta^5}{5}$ My idea was to substitute $u = \sqrt[5]{5 \cdot x} $ and then plug it into $f(x)$ and reduce it to a minimal…
user315567
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A question on minimal polynomials (with relation to some basis)

Let $\mathscr{A}$ be a linear transformation on $n$-dimensional vector space $V$. $0\neq\alpha\in V$ (which a vector field on an arbitrary field $\mathbb{F}\supset \mathbb{Q}$). Then it is easy to see that there exists a unique polynomial…
xldd
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Factorizing $X^{15}-1$ over $F_2$ using conjugacy classes

We know that $X^{15}-1=(X+1)(X^4+X+1)(X^4+X^3+X^2+X+1)(X^2+X+1)(X^4+X^3+1)$ over $F_2$ where the polynomials come from the conjugacy classes in $F_{16}$: {1}, {$a,a^2,a^4,a^8$}, {$a^3,a^6,a^9,a^{12}$}, {$a^5,a^{10}$} and…
Cecilie
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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 we have instead a block triangular…
kern711
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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 $K(\beta)$ (resp. $K(\alpha)$) are the same as the minimal…
roi_saumon
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Minimal polynomial of $\sqrt{3}$ over $\Bbb{Q}(\sqrt[6]{3})$

How to find a minimal polynomial $\sqrt{3}$ over $\Bbb{Q}(\sqrt[6]{3})$? $\Bbb{Q}(\sqrt{3}) \subset \Bbb{Q}(\sqrt[6]{3})$, because $(\sqrt[6]{3})^{3}=\sqrt{3}$ $[\Bbb{Q}(\sqrt[6]{3}): \Bbb{Q}]=[\Bbb{Q}(\sqrt{3}):\Bbb{Q}] * [\Bbb{Q}(\sqrt[6]{3}):…
Hikicianka
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Finding the minimal polynomial of $e^{2πi/5}$ over $\mathbb Q$

How would one find the minimal polynomial of $e^{2πi/5}$ over $\mathbb Q$? I have tried this: $$\text {Let } a = e^{2πi/5}$$ $$\implies a = ({e^{2πi}})^{1/5}\implies a = 1^{1/5}\implies a - 1= 0$$ Since the polynomial $a - 1$ is monic and of least…
mildog8
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