Related papers: Existence theorem of finite Krasner hyperfields

Hobby has recently shown that almost all finite hyperfields of even order fail to be the quotient of a field. Using a probabilistic argument, we extend this result to all orders: a finite hyperfield is almost always non-quotient. This…

We show that there are only finitely many finite fields whose members are the sum of an $n$-potent element and a $5$-potent element. Combining this with the algorithmic results provided by S.D. Cohen {\it et al.}, we confirm in the…

This is a presentation of explicit methods to construct higher local class field theory by using topological K-groups, explicit symbols and a generalization of Neukirch-Hazewinkel's axiomatic approaches. The existence theorem is discussed…

We show that for every $r \geq 1$, and all $r$ distinct (sufficiently large) primes $p_1,..., p_r > p_0(r)$, there exist infinitely many integers $n$ such that ${2n \choose n}$ is divisible by these primes to only low multiplicity. From a…

Smoktunowicz, Lenagan, and the second-named author recently gave an example of a nil algebra of Gelfand-Kirillov dimension at most three. Their construction requires a countable base field, however. We show that for any field $k$ and any…

The main result of this thesis is to show that there are only finitely many integers $n$ such that both $n$ and $d(n)$ are highly composite numbers at the same time, where $d(n)$ is the divisor function. Bertrand's postulate [4] is used…

Let K be a complete discretely valued field with residue field k. If char(K) = 0, char(k) = 2 and the 2-rank of k is d, we prove that there exists an integer N depending on d such that the u-invariant of any function field in one variable…

It is shown that any finitely generated subring of a global field has a universal first-order definition in its fraction field. This covers Koenigsmann's result for the ring of integers and its subsequent extensions to rings of integers in…

For integers $k \geq 2$ and $n \geq k+1$, we prove the following: If $n\cdot k$ is even, there is a connected $k$-regular graph on $n$ vertices. If $n\cdot k$ is odd, there is a connected nearly $k$-regular graph on $n$ vertices.

In this survey, we present five different proofs for the transcendence of Kempner's number, defined by the infinite series $\sum_{n=0}^{\infty} \frac{1}{2^{2^n}}$. We take the opportunity to mention some interesting ideas and methods that…

We prove a version of Garaev's sum product theorem in the set of finite fields with non-prime order. Because of the presence of subfields, this seems to require some hypotheses on the set. We work under a condition analogous to having…

We present a new structure theorem for finite fields of odd order that relates multiplicative and additive structure in an interesting way. This theorem has several applications, including an improved understanding of Dickson and Chebyshev…

We give an upper bound for the number of points of a hypersurface over a finite field that has no lines on, in terms of the dimension, the degree, and the number of the elements of the finite field.

We prove a new formula for the generating function of polynomials counting absolutely stable representations of quivers over finite fields. The case of irreducible representations is studied in more detail.

Let $n>2r>0$ be integers. We consider families $\mathcal{F}$ of subsets of an $n$-element set, in which the union of any two members has size at most $2r$. One of our results states that for $n\geq 6r$ the number of members of size…

Abhyankar proved that every field of finite transcendence degree over $\mathbb{Q}$ or over a finite field is a homomorphic image of a subring of the ring of polynomials $\mathbb{Z}[T_1, \dots, T_n]$ (for some $n$ depending on the field). We…

The first part is expository: it explains how finite fields may be used to prove theorems on infinite fields by a reduction mod p process. The second part gives a variant of P.Smith's fixed point theorem which applies in any characteristic.

Ramsey proved that for every positive integer $n$, every sufficiently large graph contains an induced $K_n$ or $\overline{K}_n$. Among the many extensions of Ramsey's Theorem there is an analogue for connected graphs: for every positive…

For a set $P$ of $n$ points in $\mathbb R^d$, for any $d\ge 2$, a hyperplane $h$ is called $k$-rich with respect to $P$ if it contains at least $k$ points of $P$. Answering and generalizing a question asked by Peyman Afshani, we show that…

Under sufficiently strong assumptions about the first term in an arithmetic progression, we prove that for any integer $a$, there are infinitely many $n\in \mathbb N$ such that for each prime factor $p|n$, we have $p-a|n-a$. This can be…