A Converse of Artin′s Density Theorem: The Case of Cubic Fields

✍ Scribed by I. Delcorso; R. Dvornicich

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 510 KB
• Volume: 45
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1993.1063

✦ Synopsis

The following problem may be considered as an inverse of Artin's density theorem: Given (n \geqslant 2) and (p) prime, does there exist a density for the set of algebraic integers (\alpha) of degree (n) for which (p) has an assigned splitting in (\mathbf{Q}(\alpha)) ? We find that such a set has a density, and we recover the density predicted by Artin's theorem when (p \rightarrow \infty). Further we given explicit formulae for all splittings in cubic fields.

(1) 1993 Academic Press. Inc