# Tagged Questions

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### Galois field splitting a polynomial

Can someone explain to me how i would go about doing a problem like this? I don't really know where to start. GF refers to a Galois field. I'm struggling to even understand exactly what they want me ...
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### Discriminant of $f(x)=x^3+ax+b$

Suppose we have the polynomial $f(x)=x^3+ax+b$, with roots $\alpha, \beta, \gamma$ in $\mathbb{C}$, and let $\Delta = (\alpha - \beta)(\beta - \gamma)(\gamma - \alpha)$. Is there any quick way of ...
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### Restrictions on the coefficients that allow a polynomial in a field of characteristic 0 to be solvable by radicals and the associated formula.

We know that a general polynomial $p(x) \in \mathcal{F} \left[ x \right]$, $\deg{ p } = n$, (char(${\mathcal{F}}) = 0$) is not solvable by radicals if $n \geq 5$. However, I was wondering what ...
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### Relation between divisibility of polynomials in different rings, $h | f$ in $\mathbb{Z}[x], \mathbb{Z}/p^k\mathbb{Z}[x]$ and $\mathbb{F}_p[x]$

Let $p$ be a prime, $k$ a positive integer. Let $f,h \in \mathbb{Z}[x]$ be polynomials such that $h | f \mod p^k$ in $(\mathbb{Z}/p^k\mathbb{Z})[x]$ $h \mod p$ is irreducible in $\mathbb{F}_p$ ...
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### Reducibility of a Cyclotomic Polynomial under the ring homomorphism $\mathbb{Z} \rightarrow \mathbb{F}_p$

I'm working through the following question: Question Reference: Oxford Part I Paper B2 2003 Find the monic polynomial $f \in \mathbb{Z}[X]$ whose roots are the complex primitive ...
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### Why do you need to introduce complex numbers in order to solve cubics by radicals?

Historically, complex numbers were first introduced (by Bombelli) in order to provide a method for solving the cubic by radicals. As far as I can tell, previous solutions of the cubic by radicals ...
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### Is the real root of $x^4+3 x-3$ constructible?

I don't know how to solve the following: Let $\alpha$ be a real root of $f(x)=x^4+3x-3\in Q[x]$. Is $\alpha$ a constructible number? Any help is welcome.
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### How does Dedekind's Theorem work for a prime dividing the discriminant of a number field?

Let $f \in \mathbb{Z}[x]$ be an irreducible monic polynomial, let $N$ be its splitting field, and let $G$ be the Galois group of the extension $N/\mathbb{Q}$. Let $p$ be a prime dividing the ...
What is the minimal polynomial over $\Bbb Q[x]$ of \alpha[k] = n\cdot ...