• Corpus ID: 220793933

Hilbert's Tenth Problem and the Inverse Galois Problem

@article{Balestrieri2020HilbertsTP,
  title={Hilbert's Tenth Problem and the Inverse Galois Problem},
  author={Francesca Balestrieri and Jennifer Park and Alexandra Shlapentokh},
  journal={arXiv: Number Theory},
  year={2020}
}
We show that the inverse Galois problem over global fields can be reduced to Hilbert's tenth problem over $\mathbb{Q}$ (and consequently over $\mathbb{Z}$); that is, if there is an algorithm to decide the existence of a solution for an arbitrary polynomial equation over $\mathbb{Q}$, then there is an algorithm to decide whether a certain group $G$ appears as the Galois group of a finite extension of a given global field $K$. 

References

SHOWING 1-10 OF 39 REFERENCES
Towards Hilbert’s tenth problem for rings of integers through Iwasawa theory and Heegner points
For a positive proportion of primes p and q , we prove that $${\mathbb {Z}}$$ Z is Diophantine in the ring of integers of $${\mathbb {Q}}(\root 3 \of {p},\sqrt{-q})$$ Q ( p 3 , - q ) . This provides
Diophantine definability over some rings of algebraic numbers with infinite number of primes allowed in the denominator
Abstract. Let K be a number field. Let W be a set of non-archimedean primes of K, let OK, W={x∈K∣ordpx≥0∀p∉W}. Then if K is a totally real non-trivial cyclic extension of ℚ, there exists an infinite
Hilbert’s tenth problem for a class of rings of algebraic integers
We show that Z is diophantine over the ring of algebraic integers in any number field with exactly two nonreal embeddings into C of degree > 3 over Q. Introduction. Let R be a ring. A set S c Rm is
Embedding problems over large fields
In this paper we study Galois theoretic properties of a large class of fields, a class which includes all fields satisfying a universal local-global principle for the existence of rational points on
DIOPHANTINE DEFINABILITY OVER HOLOMORPHY RINGS OF ALGEBRAIC FUNCTION FIELDS WITH INFINITE NUMBER OF PRIMES ALLOWED AS POLES
Let K be an algebraic function field over a finite field of constants of characteristic greater than 2. Let W be a set of non-archimedean primes of K, let . Then for any finite set S of primes of K
Extension of Hilbert's tenth problem to some algebraic number fields
We extend the solution of Hilbert's tenth problem to algebraic number fields having one pair of complex conjugated embeddings. The proof is based on the extended method of J. Denef used for totally
...
...