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A Density of Ramified Primes.
Stephanie Chan1, Christine McMeekin2, Djordjo Milovic1
1Department of Mathematics, University College London, London, UK.
Research in Number Theory
|November 22, 2021
Summary
Researchers studied number fields, defining a new family indexed by rational primes. They found the density of primes with specific ramified factorizations is between 0 and 1, offering an explicit formula.
Area of Science:
- Algebraic Number Theory
- Analytic Number Theory
Background:
- Cyclic number fields with specific properties (odd degree, odd narrow class number, inert 2) are foundational.
- Understanding prime ideal factorization in extensions is crucial for number theory research.
Purpose of the Study:
- To define and analyze a new family of number fields, denoted .
- To investigate the density of rational primes (p) that split completely in an initial field (K) and ramify with degree 2 in the new family.
- To provide an explicit formula for this density, conditional on a conjecture about short character sums.
Main Methods:
- Construction of a family of number fields based on a base cyclic field and rational primes .
- Analysis of the ramification behavior of primes in the constructed fields.
- Application of a standard conjecture on short character sums to determine prime densities.
- Detailed study of the joint distribution of spins of prime ideals.
Main Results:
- A formula is derived for the density of rational primes exhibiting specific ramified factorizations in .
- This density is proven to be strictly between 0 and 1, conditional on the short character sums conjecture.
- The results are unconditional for the cubic case (degree 3 extensions).
Conclusions:
- The study provides new insights into the distribution of prime numbers in specific algebraic structures.
- The explicit formula for prime density offers a quantitative tool for further number theoretic investigations.
- The findings highlight the importance of prime ideal distribution in understanding number field properties.
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