Intereting Posts

The space $C$ with $\|f\|:=\int_a^b |f(x)|d(x)$ is a normed space but not a Banach space
Number of limit points of a continued exponential
Prove that $f$ has derivatives of all orders at $x=0$, and that $f^{(n)}(0)=0$ for $n=1,2,3\ldots$
How can I find the number of the shortest paths between two points on a 2D lattice grid?
Prove that : $|f(b)-f(a)|\geqslant (b-a) \sqrt{f'(a) f'(b)}$ with $(a,b) \in \mathbb{R}^{2}$
Has the polynomial distinct roots? How can I prove it?
Proving left-invariance (and proof-verification for right-invariance) for metric constructed from left-invariant Haar measure
Number of queries required to find the function.
interior points and convexity
Derivative of Quadratic Form
Wisdom of using Sympy as a first CAS
Absolute value in trigonometric substitutions
Minimal polynomial of the operator $T:V\oplus W\to V\oplus W$
Why use the derivative and not the symmetric derivative?
Why does my professor say that writing $\int \frac 1x \mathrm{d}x = \ln|x| + C$ is wrong?

Inspired by this question, I was wondering about the following problem.

$\alpha,\beta,\gamma,\ldots$ are the roots of an irreducible

polynomial over $\mathbb{Q}$. How to compute the coefficients of $$

(x-\alpha\beta)(x-\alpha\gamma)\cdot \ldots\cdot (x-\beta\gamma)\cdot\ldots, $$ i.e. the candidate minimal polynomial

of $\alpha\beta$, in the most efficient way?

My approach is based on a simple lemma and a general fact. If $p_m$ is the power sum $\alpha^m+\beta^m+\gamma^m+\ldots$ and $e_i$ is the $i$-th elementary symmetric polynomial, the identity

$$ \exp\left(-\sum_{m\geq 1}\frac{p_m}{m}x^m\right) = \sum_{r\geq 0}(-1)^r e_r\,x^r$$

gives a way to compute $p_1,p_2,p_3,\ldots$ from $e_1,e_2,e_3,\ldots$ through the Taylor series of a logarithm ($\text{LOG}$) as well as $e_1,e_2,e_3,\ldots$ from $p_1,p_2,p_3,\ldots$ through the Taylor series of an exponential ($\text{EXP}$). Moreover, the characteristic polynomial of the sequence $\{p_k\}_{k\geq 0}$ is exactly the minimal polynomial of $\alpha$, hence we may compute $p_{n+1},p_{n+2},\ldots,p_{n+m}$ from $p_1,p_2,\ldots,p_n$ with a simple recursive approach (I will call this procedure $\text{CH}$, from Cayley-Hamilton). $\text{L}$ will be my previously mentioned lemma, i.e.

$$ p_k(\alpha\beta,\alpha\gamma,\ldots) = \frac{1}{2}\left(p_k^2-p_{2k}\right).$$

Now my algorithm goes as follows:

- $H^1$ of $\Bbb Z$ as a trivial $G$-module is the abelianization of $G$
- Conjugate class in the dihedral group
- Find the degree of the splitting field of $x^4 + 1$ over $\mathbb{Q}$
- Show that $\mathbb{Q}(\sqrt{2}+\sqrt{5})=\mathbb{Q}(\sqrt{2},\sqrt{5})$
- Every Submodule of a Free Module is Isomorphic to a Direct Sum of Ideals
- Group presentation for semidirect products

(1). $$ e_i(\alpha,\beta,\gamma,\ldots)_{i\leq n}\quad\xrightarrow{\text{LOG}}\quad p_i(\alpha,\beta,\gamma,\ldots)_{i\leq n}$$

(2). $$ p_i(\alpha,\beta,\gamma,\ldots)_{i\leq n}\quad\xrightarrow{\text{CH}}\quad p_i(\alpha,\beta,\gamma,\ldots)_{i\leq n(n-1)}$$

(3). $$ p_i(\alpha,\beta,\gamma,\ldots)_{i\leq n(n-1)}\quad\xrightarrow{\text{L}}\quad p_i(\alpha\beta,\alpha\gamma,\ldots)_{i\leq \binom{n}{2}}$$

(4). $$ p_i(\alpha\beta,\alpha\gamma,\ldots)_{i\leq \binom{n}{2}}\quad\xrightarrow{\text{EXP}}\quad e_i(\alpha\beta,\alpha\gamma,\ldots)_{i\leq \binom{n}{2}}.$$

- Why are fields with characteristic 2 so pathological?
- Let $G$ a group of order $6$. Prove that $G \cong \Bbb Z /6 \Bbb Z$ or $G \cong S_3$.
- Can we permute the coefficients of a polynomial so that it has NO real roots?
- Is the localization of a PID a PID?
- There are at least three mutually non-isomorphic rings with $4$ elements?
- Why is $\mathbb{Q}(t,\sqrt{t^3-t})$ not a purely transcendental extension of $\mathbb{Q}$?
- Can a smooth function on the reals form a non-commutative semigroup?
- $\mathbb{F}_p/(X^2+X+1)$ is a field iff $p \equiv 2 \bmod 3$
- Proving $2,3,1+\sqrt{-5}$ and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}$
- Proof that all abelian simple groups are cyclic groups of prime order

I don’t see that the assumption that $\alpha$ and $\beta$ are roots of the same polynomial is particularly helpful. Suppose they have minimal polynomials $f(\alpha), g(\beta)$. Write down the companion matrices of these polynomials, then the Kronecker product of these matrices, then compute the characteristic polynomial of the Kronecker product. This polynomial isn’t guaranteed to be irreducible, but from here you can factor it.

- Linear Algebra, Vector Space: how to find intersection of two subspaces ?
- Any nonabelian group of order $6$ is isomorphic to $S_3$?
- Show that a function from a Riemann Surface $g:Y\to\mathbb{C}$ is holomorphic iff its composition with a proper holomorphic map is holomorphic.
- Probability of combinations of beads on cut necklaces (mass spectrometry physics problem)
- Integration method for $\int_0^\infty\frac{x}{(e^x-1)(x^2+(2\pi)^2)^2}dx=\frac{1}{96} – \frac{3}{32\pi^2}.$
- Why is $\frac{d^n}{dx^n}(y(x))$ the notation for the $n$th derivative of $y(x)$, instead of $\frac{d^n}{d^nx}(y(x))$?
- Chain Rule Intuition
- Explaining the product of two ideals
- Prove the inequality for composite numbers
- Limit involving $(\sin x) /x -\cos x $ and $(e^{2x}-1)/(2x)$, without l'Hôpital
- Find the Green's Function and solution of a heat equation on the half line
- proof that an alternative definition of limit is equivalent to the usual one
- analytic solution to definte integral
- Finding a correspondence between $\{0,1\}^A$ and $\mathcal P(A)$
- Number of finite simple groups of given order is at most $2$ – is a classification-free proof possible?