Unique factorization for the semigroup generated by {2cos(π/n) | n>3}?

Let S be the multiplicative semigroup of numbers generated by B={2cos(πn)∣n≥4}. Question: Does every number of S factorize uniquely (up to perm.) as a product of elements in B? Note that 2cos(πn)=eiπn+e−iπn, so we can reformulate the question as follows. Let 4≤n1≤⋯≤nr and 4≤m1≤⋯≤ms such that ∑(ϵ1,…,ϵr)∈{±1}reiπ(∑rk=1ϵknk)=∑(ϵ1,…,ϵs)∈{±1}seiπ(∑sk=1ϵkmk) Then, is it true that r=s and nk=mk,∀k? … Read more

Polynomial equations in many variables have solutions (Lang 1952 paper)

I am trying to understand the proof of the following result: Suppose F is a function field in k variables over an algebraically closed field. Let f1,…,fr∈F[x1,…,xn] be polynomials without constant term of degrees d1,…,dr, respectively. Assume that n>dk1+…+dkr. Then f1,…,fr have a non-trivial common zero. If F is instead a function field in k … Read more

Cyclotomic Extension of a Perfectoid Space

Maybe, I am being stupid, but when I consider ramified extension of a perfectoid field with the characteristic 0, I cannot find the correspondent field with characteristic p. Let me put it more precisely. Let K be the completed field of Qp(p−∞). Consider ζp be a pth root of unity, and let L=K(ζp). According to … Read more

On the quadratic equivalence of fields

I have spent the past two years studying abstract Witt rings. These objects are a generalization of “The Witt ring of a field,” an algebraic invariant of fields of characteristic not equal to 2. Classic examples are W(R)=Z and W(C)=Z/2Z. The study of these Witt rings of fields has produced a family of abstract Witt … Read more

Field K and integer n such that the etale fundamental group of (GL_n)_K is the profinite completion of GL_n(K)?

Prove or disprove: Claim: there exists a field K and an integer n such that πet1((GLn)K) is isomorphic to the profinite completion of the abstract group GLn(K)? Note that then GK≅^GLn(K)/ˆZ. In particular, GK must have a (topological) generating set of size the cardinality of K. Answer AttributionSource : Link , Question Author : David … Read more

Issue with “definition” of pseudo algebraically closed fields

I’m having an issue with a sentence in Chapter 11 of Fried & Jarden’s Field Arithmetic. As a “motto” for pseudo algebraically closed (PAC) fields, they say they are fields K such that “each nonempty variety defined over K has a K-rational point”. No mention of absolute irreducibility is made at this point. My issue … Read more

Products of short elements in a field

Consider a field $F$ of characteristic zero. Let $L=F[\alpha]$ be an extension of degree $d.$ We call an element $$ x=x_0 + x_1 \alpha +\ldots+ x_{d-1}\alpha^{d-1}\in L $$ short if $x_{d-1}=0.$ Under which conditions on $\alpha$ every element in $L^\times$ is a product of short elements? It is easy to see that every element of … Read more

Existence of an irreducible polynomial that does not divide xn+ax^n + a

The question: Can one characterize all fields K over which there exists an irreducible polynomial f(x) that does not divide a polynomial of the form xn+a? Examples: Such a polynomial clearly exists over Q. It also exists over R (the polynomial with roots (3+4i)/5 and (3−4i)/5). As far as I understand, such polynomials do not … Read more

Minimum product of degrees of generators of finite field extension

Suppose L/K is a finite extension of fields. When is it true that min By the primitive element theorem, this is certainly true if L/K is separable. This paper seems to suggest the condition does not hold in general for purely inseparable extensions. But I cannot find or think of an explicit counterexample; does anybody … Read more