skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


  • Home
  • Search Results
  • Page 1 of 1

Search for: All records

Creators/Authors contains: "Shlapentokh, Alexandra"

Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher. Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?

Some links on this page may take you to non-federal websites. Their policies may differ from this site.

  1. ; ; (, Journal of the London Mathematical Society)
    Abstract We introduce a notion of algorithmic randomness for algebraic fields. We prove the existence of a continuum of algebraic extensions of that are random according to our definition. We show that there are noncomputable algebraic fields which are not random. We also partially characterize the index set, relative to an oracle, of the set of random algebraic fields computable relative to that oracle. In order to carry out this investigation of randomness for fields, we develop computability in the context of the infinite Galois theory (where the relevant Galois groups are uncountable), including definitions of computable and computably enumerable Galois groups and computability of Haar measure on the Galois groups. 
    more » « less
    Full Text Available 
  2. ; ; ; ; (, Notices of the American Mathematical Society)
    Full Text Available 
  3. ; ; (, Journal of Number Theory)
    Full Text Available 
  4. ; ; (, Acta Arithmetica)
    We consider first-order definability and decidability questions over rings of integers of algebraic extensions of $$\Q$$, paying attention to the uniformity of definitions. The uniformity follows from the simplicity of our first-order definition of $$\Z$$. Namely, we prove that for a large collection of algebraic extensions $$K/\Q$$, $$ \{x \in \oo_K : \text{$$\forall \e \in \oo_K^\times \;\exists \delta \in \oo_K^\times$ such that $$\delta-1 \equiv (\e-1)x \pmod{(\e-1)^2}$$}\} = \Z $$ where $$\oo_K$$ denotes the ring of integers of $$K$$. One of the corollaries of our results is undecidability of the field of constructible numbers, a question posed by Tarski in 1948. \end{abstract} 
    more » « less
    Full Text Available