All Questions
Tagged with gr.group-theory nt.number-theory
398 questions
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Existence of Finite Amicable Groups
I'm interested in exploring the concept of "amicable groups" as follows:
Definition. Two finite groups $G$ and $H$ are called amicable groups if:
$G$ is the direct sum of proper subgroups ...
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0
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89
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Which elements in $\mathrm{Aut}(\widehat{F_2})$ preserve the procyclic subgroup generated by the commutator $c=[a,b]$?
Let $F_2$ denote the free group over two generators $a,b$, and we denote $c=[a,b]$ as the commutator. It is well-known that any automorphism $\psi$ of $F_2$ preserves the conjugacy class of the ...
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2
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343
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Non-commuting elements of finite orders in a reductive group over a p-adic field
Let $k$ be a $p$-adic field and $G$ be a connected non-abelian reductive algebraic group over $k$. I am asking for a proof of the following lemma:
Lemma. Assuming that $p$ is "good" for $G$,...
- 14.1k
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65
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Higher-order obstructions in thin group orbits
Let $G$ be a finitely generated group acting on the integers $\mathbb{Z}$. Let $O_a = \{g \cdot a : g \in G\}$ be the orbit of an integer $a$ under this action. Assume that $O_a$ is a thin orbit, ...
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86
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Is the Galois closure of a $p$-adic Lie group extension also a $p$-adic Lie group extension?
Let $p$ be a prime. Let $K$ be a number field and $L/K$ be an infinite extension which is not necessarily Galois. Suppose that the automorphism group $\text{Aut}(L/K)$ of $L/K$ is $p$-adic analytic, i....
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258
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Density of numbers where a large prime factor satisfies a congruence
I am looking for an upper bound on the number of integers $n<x$ such that $n$ has a prime factor $p>\log(x)^{(1+\delta)}$ such that $p \equiv a \mod b$. Where $a,b$ are fixed and coprime and $0&...
CommunityBot
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Prime index subgroups of $\langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle$ that is invariant under matrix $Q$
Let $Q $ be a matrix in $ \operatorname{GL}(2, \mathbb{Q}) $ and consider the group
$G = \langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle := \langle Q^{i}(v) \mid i \in \mathbb Z, v \in \...
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Is every (finite-dimensional, complex) representation of a finite group defined over the algebraic integers?
Is every (finite-dimensional, complex) representation of a finite group defined over the algebraic integers?
Apologies in advance if this is obvious.
Edit, 5/31/24: Since this question is getting some ...
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The existence of solutions to linear systems of equations over the integer ring $\mathbb{Z}$
There are already detailed results on the solutions of linear equations over fields, but I'd like to inquire about any good conclusions regarding the solutions of linear equations over the integer ...
- 30.1k
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Bourgain-Gamburd-like theorems in the non-algebraic case
For $\mu$ a Borel probability measure on the compact group $G=\operatorname{SU}(d)$, Bourgain-Gamburd prove that the spectral radius of the associated operator on $L^2(G)$ is strictly less than one, ...
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Subgroup of p-adic units
Let $\left(\widehat{\mathbb Z}\right)^\times=\prod_p{\mathbb Z}_p^\times$
be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$.
We give it the product ...
- 460
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98
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Estimating the cardinality of the set of conjugacy classes of subgroups in a finite group of given order
1. Let $G$ be a finite group of order $n$. I need an estimate for the number $c$ of conjugacy classes of subgroups $D\subseteq G$.
Note that any subgroup of $G$ contains $1_G$, and so the set of all ...
- 14.1k
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Subgroups of ${\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$
Assume that $n \geq 3$. Is there a subgroup $H \leq {\rm GL}_{2}(\mathbb{Z}/2^{n} \mathbb{Z})$ whose order is a power of $2$ so that
$\bullet$ $\det : H \to (\mathbb{Z}/2^{n} \mathbb{Z})^{\times}$ is ...
- 20.4k
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1k
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Group generated by two irrational plane rotations
What groups can arise as being generated by two rotations in $\mathbb R^2$ by angles $\not \in \mathbb Q\pi$?
If the centers of the rotations coincide, then the rotations commute and generate some ...
- 63.9k
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246
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How do "Kummer closures" of fields look?
Let $F$ be a field and $A$ a finite abelian group. You can ask: does the regular representation $F[A]$ of $A$ split as a direct sum of 1-dimensional representations? This is equivalent to the ...
- 148k
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232
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Transfer for the group of coinvariants: a reference request
Let $G$ be a group and $M$ be a $G$-module,
that is, an abelian group written additively on which $G$ acts:
$$ (g,m)\mapsto g m.$$
We consider the group of coinvariants
$$ M_G:=G/\langle g m -m\ |\ g\...
- 5,898
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1
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252
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Which algebraic groups are generated by (lifts of) reflections?
$\DeclareMathOperator\SL{SL}$The Cartan–Dieudonné theorem
states that each element $g \in \operatorname{O}(V)$, where $V$ is a quadratic space of dimension $n$ over a field of characteristic $\neq 2$, ...
- 63.9k
7
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201
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Lifting SL2(k) to a subgroup of Witt vectors
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\W{W}$Let $k$ be a finite field, and let $\W_n(k)$ be the degree $n$ Witt vectors over $k$ (so $\W_1(k) = k$).
Does there ...
- 63.9k
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For which subgroups the transfer map kills a given element of a group?
$\newcommand{\ab}{{\rm ab}}
\newcommand{\ord}{{\rm ord}}
$Let $G$ be a finite or profinite group. Consider the abelianized group
$$G^\ab=G/G'$$
where $G'$ is the commutator subgroup of $G$.
Let $H\...
- 1,907
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Navigating $\mathbb{Z}/p\mathbb{Z}$
$\newcommand{\Z}{\mathbb{Z}}$Let's consider a silly-looking question first. Consider $\Z/p\Z$. Say I am allowed the two operations $x\mapsto x+1$ and $x\mapsto 2x$. Then, starting from $0$, I can ...
- 63.9k
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646
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Are these two methods for constructing Hadamard matrices known?
These two observations came while researching the empty set of odd perfect numbers and unitary perfect numbers:
Context:
Let $n$ be a natural number and $D_n$ be the set of divisors.
We can make this ...
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3
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345
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A subgroup of index 2 for a solvable finite group transitively acting on a finite set of cardinality $2m$ where $m$ is odd
Let $G$ be a finite group acting transitively on the finite set $X=\{1,2,\dots, 2m\}$
of cardinality $2m\ge 6$ where $m$ is odd.
Question 1. Is it true that $G$ always has a subgroup $H$ of index 2
...
- 14.1k
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169
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Square hidden number problem
Suppose I have a mystery number $m$ modulo $p$ that I wish to find. I know the value of $m+x_i^2$ where $x_i$ is randomly chosen modulo $p$ for some large number of different $x_i$, $N$ many, $N \gg \...
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71
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Bias of $a^k / q$ modulo $q$?
Let $q$ be a prime. Let $0< a < q$ be an integer so that it is primitive modulo $q$. Let $k$ be a random integer up to $q-1$. Consider
$$a^k = b_k + q * c_k$$
as $k$ varies modulo $q^2$. So $b_k$...
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175
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Intrinsic maps between complex integers modulo $p$ and integers modulo $p+2$
$\DeclareMathOperator\GF{GF}$Let $p$ and $p+2$ be twin primes. Let's assume that $-1$ is not a quadratic residue modulo $p$ (and therefore is a Q.R. modulo $p+2$).
Consider the complex numbers $a+bi$ ...
- 63.9k
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195
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Trying to solve for the remainder of $a^q$ modulo $q$
Let $q$ be a prime and $a$ be a number from $0$ to $q-1$ (not an equivalence class).
The elements $a^q$ are exactly the elements of order $q-1$ modulo $q^2$.
I'm trying to solve the equation:
$$a+2*\...
- 34.3k
7
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633
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Given a rational matrix $Q$, can we generate $\langle Q^{i}(v)\mid i\in\mathbb Z,v\in\mathbb Z^{2}\rangle$ using only non-negative powers of a matrix?
I have copied this question from StackExchange, thank you to those who helped me to improve the question. (apology if you have seen this question already)
Let $Q $ be a matrix in $ \operatorname{GL}(...
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278
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Which pairs of conjugates of $\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right)$ generate $\operatorname{SL}(2,\mathbb{Z})$?
When do two distinct conjugates of $U := \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ generate $\DeclareMathOperator\SL{SL}\SL(2,\mathbb{Z})$? The classic example is $U,L^{-1}$, where $L = \...
- 15.4k
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73
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Is the discrete logarithm equivalent to solving polynomial discrete logarithms?
Suppose we can quickly solve the discrete logarithm modulo $p$. Let's say $2$ is a generator so we can quickly find $l$ for which $2^l =h$ for any given target $h$.
An interesting observation is that ...
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2k
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Trivial homomorphism from a non-abelian group to an abelian group
I am stuck on this problem and cannot seem to find a good reasoning for drawing the required conclusion. The problem is as follows:
Let $m\in \mathbb{N}$ and $n>3$. I want to show that there can be ...
- 24.2k
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Reduction mod 2 for orthogonal groups
Setting Let $k$ be a real quadratic field, $\mathbb Z_k$ its ring of integers. Let $n$ be an even integer $A$ a symmetric $n$-by-$n$ matrix with coefficients in $\mathbb Z_k$. Let $L$ be the lattice $\...
- 63.9k
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3
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457
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Intersection of Fourier analysis (especially on the transform) and group theory, number theory, dynamical systems, etc
I am considering a PhD research topic. I only have a math Bachelor's degree with working experience mostly in teaching and I have been working on a paper. I have deep interest in Fourier Series and ...
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Showing non-expansion for $x\rightarrow x+1, x\rightarrow 2x.$
Construct a graph having $V=\mathbb{Z}/p\mathbb{Z}$ as its set of vertices and $\{\{x,x+1\}: x \in V\}\cup \{\{x,2x\}: x \in V\}$ as its set of edges. This graph is not an expander - quite ...
- 4,713
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Elements living in the conjugacy class and in the centralizer of an $m$-cycle in $A_m$
Let $m>1$ be an odd natural number, $x$ a $m$-cycle in $A_m$, the alternating group in $m$ letters, $C$ the conjugacy class of $x$ in $A_m$.
Question: How can I describe the elements in the set $\{ ...
- 6,367
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Some necessary condition for $\gcd(m,n) $ be a proper divisor of $\gcd(mk_2 +nk_1,mn) $ [closed]
Let $m,n,k_1,k_2 $ be natural numbers such that $(k_1,m)=(k_2,n)=1 $.
Statement 1: $\gcd(m,n) $ is a proper divisor of $\gcd(mk_2 +nk_1,mn) $, for every $k_1,k_2$ having the above property.
Statement ...
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130
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Can we say anything about the discrete logarithm of $x+1$?
Consider the multiplicative group $\mathbb{Z} / p\mathbb{Z}$. Let $g$ be a generator and suppose $g^n = x$. Can we say anything at all about the discrete logarithm of $x+1$? That is, can we write $m$ ...
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Diameter of the unimodular group with Gauss moves
$\DeclareMathOperator\GL{GL}$Consider the unimodular group $\GL_n(\mathbb{Z})$, consisting of integral matrices $A \in \mathbb{Z}^{n \times n}$ such that that $\det(A) =\pm 1$.
It is well known that ...
- 63.9k
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145
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What is the periodicity of $((a^n \text{ modulo } p) \text{ modulo } q)$
This feels like it should be elementary but it came up in my research and I was not able to solve it.
We can ask this question for any $p$ and $q$ but,let $p$ and $q$ be primes for simplicity. The ...
- 591
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252
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Smith normal form and last invariant factor of certain matrices
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts can provide some relevant insight.
Suppose we have row vectors $x_1$, $x_2$ , $y_1$ , $y_2 ...
- 5,409
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189
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Cyclic numbers of the form $2^n + 1$
A cyclic number (or cyclic order) is a number $m$ such that the only group of order $m$ is the cyclic group $\mathbb{Z}/m\mathbb{Z}$. The set of cyclic numbers admits a couple of cute number-theoretic ...
- 756
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Sparsity of q in groups PSL(2,q) that are K_4-simple
One of the problems that has come up during my research concerns $K_4$-simple groups (simple groups with $4$ prime divisors). The only (potentially) infinite family of groups satisfying this condition ...
- 44.4k
1
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98
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Existence of countable dense normal subgroups of global Galois group
Let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group of the maximal extension of $ K $ (inside a fixed algebraic closure of $K$) unramified outside $ S $. In ...
- 63.9k
12
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1
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450
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abelian quotients of permutation groups
Let $G$ be a subgroup of the permutation group $S_n$, and let $H$ be a normal subgroup of $G$ such that the quotient group $G/H$ is abelian. What is the best known upper estimate for the cardinality $...
- 63.9k
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447
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Calculating the centralizer of a subgroup of $\mathrm{GL}(n, \mathbb{Z})$
I want to calculate the centralizer of a subgroup of $\mathrm{GL}(n, \mathbb{Z})$ when its generators $G_i$ (whose number is finite) are known.
For any matrix $S$ that commutes with the group: $G_iS$ =...
- 37.4k
8
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1
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255
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Automorphisms of Lubotzky–Phillips–Sarnak graphs
For the Lubotzky–Phillips–Sarnak (LPS) graph $X^{p,q}$, what is its automorphism group? These graphs are not just ($p+1$)-regular but are Cayley graphs for $G=\mathrm{PSL}_2(\mathbb{F}_q)$, so clearly ...
- 12.4k
5
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1
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513
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Learning Inverse Galois Theory
Can someone give me a roadmap for learning Inverse Galois theory?
I am a PhD student in the representation theory of finite groups. I studied Galois theory when I was an undergraduate student. The ...
- 16.2k
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6
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2k
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Lifting units from modulus n to modulus mn.
Background
In his beautifully short answer to a previous question of mine, Robin Chapman asserted the following.
Let $m,n,r$ be natural numbers with $r$ coprime to $n$. Then there is $r' \equiv r ...
- 33.8k
3
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1
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384
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Using the Lehmer quintic to solve $11$-degree equations and higher?
(This is a natural continuation of a previous post.)
I. Quintic method
Given the Lehmer quintic,
$$x^5 + n^2x^4 - (2n^3 + 6n^2 + 10n + 10)x^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)x^2 + (n^3 + 4n^2 + 10n + ...
- 79.9k
4
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1
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243
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Existence of intermediate field extensions for tamely ramified p-adic extensions
Let $p$ be a prime, and let $K/\mathbb{Q}_p$ be a tamely ramified finite extension of degree $n$. Let $q$ be a prime factor of $n$ with $q\neq p$. Must there exist an intermediate extension $L$ (...
- 2,959
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2
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439
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A method to generate solvable equations of degrees $p = 7, 13, 19, 31, 37,\dots$ using only cubics
I've always wondered if the DeMoivre method to generate an algebraic number $x_p$,
$$x_p = u_1^{1/p}+u_2^{1/p}$$
of degree $p$ using only quadratic roots $u_i$ could be generalized using cubic roots $...
- 12.6k