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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Simplify into reduced form
Simplify into reduced form the expression
$$\frac{1+2α+3α^2}
{(1+α)^{15}}
$$
, where the
minimal polynomial of $α ∈ \mathbb{C}$ over $\mathbb{Q}$ is $x^3 + x + 1$.
My Attempt: First, we can write $\...
5
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The Mahler measure of a integral polynomial is an algebraic integer
Let $f(X) = a_nX^n+\cdots+a_0 \in \mathbb{Z}[X]$ be a polynomial with roots $\alpha_1,\dots,\alpha_n$. The Mahler measure of $f$ is defined to be
$$
M(f) = |a_n|\prod_{i=1}^{n}\max\{1,|\alpha_i|\}.
$$
...
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Show that the minimal polynomial of $\sqrt{3}+\sqrt{5}$ over $\mathbb{Q}$ is irreducible in $\mathbb{Z}[t]$ but becomes reducible modulo any prime $p$
Problem: Find the minimal polynomial of $\sqrt{3}+\sqrt{5}$ over $\mathbb{Q}$ and show that it is an irreducible polynomial in $\mathbb{Z}[t]$ which becomes reducible modulo any prime $p$.
I found ...
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Matrix polynomials and quotient field
Let $k$ be a field and $V$ an $n$-dim $k$-vector space. Then, for every nonzero $A\in L(V),ドル the evaluation map $EV_A:k[X]\to L(V),ドル by $f(x)\mapsto f(A)$ is a unital ring homomorphism. The kernel of ...
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How can I find the minimal polynomial over a finite field in a different way?
Let $F$ be the splitting field of $f = x^{2}+x+1 \in \mathbb{F}_{5}[x]$ and let $a \in F$ be a root of $f$. Prove that the polynomial $g = Irr(a+1, \mathbb{F}_{5})$ verifies $g(a) = 3a$.
We first show ...
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Is there a minimal-degree integer polynomial $f(x)$ such that $f(\cos(2\pi/13)) = \sqrt{13}$?
I’m studying the algebraic structure of $\cos(2\pi/13),ドル which is known to be an algebraic number of degree 6ドル$ over $\mathbb{Q}$. It’s also known from Gaussian period theory that:
$$
\sqrt{13} = -2 \...
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Find the minimal polynomial of $\sqrt{13+6\sqrt{2}}\in\mathbb{R}$ over $\mathbb{Q}$
Problem: Find the minimal polynomial of $t=\sqrt{13+6\sqrt{2}}\in\mathbb{R}$ over $\mathbb{Q}$.
I found that $f(x)=x^4-26x^2+97\in\mathbb{Q}[x]$ satisfies $f(t)=0$. Now we need to show that $f$ is ...
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Regarding minimal polynomial at a vector: $\mu_{\mathcal{A},v}=\mu_{\mathcal{A}}$ over finite fields?
Here is a linear algebra exercise. I wonder can we remove the condition '$|k|=\infty$'?
Let $|k|=\infty,ドル $V$ be an $n$-dimensional $k$-vector space. For $\mathcal{A}\in \mathcal{L}(V)$ and $v\in V,ドル ...