# Tagged Questions

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### Solutions of $x^d=1$ in a finite field

Let's consider the polynomial $x^d-1$. Theory tells us that it can have at most $d$ roots in (any extension of) a given field. Here's my problem: let $A$ be the vector space spanned by ...
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### Proving $(φ(x)\cdot ψ(x)) \cdot ω(x)=φ(x) \cdot (ψ(x)\cdot ω(x))$ where $φ,ψ,ω$ are polynomials on a ring $R[X]$

If I take $3$ random polynomials $φ,ψ,ω$ on a ring $R[x]$, I'm trying to prove associativity which is very obvious. But I have trouble on the algebra part with the sums. I know that given $2$ ...
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### How to prove how many ireducible polynomials are in a polynomial ring over a finite field.

Can someone help me out with some problem concerning the cardinaliy of ireducible polynomials in $(\mathbb Z/p\mathbb Z)[x]$? Here first why the problem appeared, I want to prove that ...
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### If $[E:F]$ is finite and $\alpha \in E$ then there is an irr. polynomial in $F[x]$ with root $\alpha$

I'm studying for an exam and encountered a confusing proof of the following fact in my notes: Let $[E:F]$ be finite and $\alpha \in E$ then there is an irreducible polynomial $p(x) \in F[x]$ with ...
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### For an irreducible polynomial $p$ and root $\alpha$, $[F[\alpha]:F]$ = degree of $p$

I'm studying for an exam, and I couldn't find the proof for the following theorem in my notes: For $p(x) \in F[x]$ irreducible and $\alpha$ a root of $p$ in some extension field, $[F[\alpha]:F] =$ ...
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### Problems with a ring isomorphism

Let $k$ be a field and consider $a=(a_0,\ldots,a_n)\in k^{n+1}$ with $a_0\neq0$. Now $\rho(a)=\left(\{a_iT_j-a_jT_i\;:\; 0\le i<j\le n\}\right)$ is an homogeneous ideal of $k[T_0,\ldots,T_n]$ and I ...
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### Multiplicative Monoid, Ideal, and Cone [closed]

Let f(x) be polynomial function, i.e. a linear summation of monomials with real constant. Let ${\cal M}$ be Multiplicative Monoid. What is ${\cal M}(f(x))$?
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### Algorithms for factoring multivariate polynomials

I am wondering if there are any algorithms to factor polynomials in multiple variables, when you know that the factors are other polynomials with rational or integer coefficients. I know you have the ...
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### Solving for Coefficients of Multivariable Polynomial from Expanded Form

I have 3 single variable polynomials in x, y, and z that have been multiplied together (expanded). I have the numerical coefficients of the expanded polynomial and want to solve for the a, b, and c ...
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### Computing univariate resultant via modified Euclidean algorithm

In an answer to the question Resultant of Two Univariate Polynomials, a PDF of course slides was linked which describes a modification of Euclid's algorithm for computing univariate polynomial ...
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### positive integer polynomial under the usual polynomial multiplication

consider the set of polynomials with positive integer coefficients together with the operation, usual multiplication of polynomials. now my first question is does this set together with the ...
Let $A^n$ be an affine space over $\mathbb{C}$ and let $\mathbb{C}[X_1,\cdots,X_n]$ be the polynomial ring of $n$ variables. Then $A^n\to (\mathbb{C}[X_1,\cdots,X_n])^*$ by evaluation homomorphism, ...