𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Abelian groups and quadratic residues in weak arithmetic

✍ Scribed by Emil Jeřábek

Publisher
John Wiley and Sons
Year
2010
Tongue
English
Weight
282 KB
Volume
56
Category
Article
ISSN
0044-3050

No coin nor oath required. For personal study only.

✦ Synopsis

We investigate the provability of some properties of abelian groups and quadratic residues in variants of bounded arithmetic. Speci cally, we show that the structure theorem for nite abelian groups is provable in

), and use it to derive Fermat's little theorem and Euler's criterion for the Legendre symbol in S 2 2 + iWPHP(PV) extended by the pigeonhole principle PHP(PV). We prove the quadratic reciprocity theorem (including the supplementary laws) in the arithmetic theories T 0 2 + Count2(PV) and IΔ0 + Count2(Δ0) with modulo-2 counting principles.

📜 SIMILAR VOLUMES

Circuits in cayley digraphs of finite ab
Circuits in cayley digraphs of finite abelian groups
✍ Anne Marie Wilkinson 📂 Article 📅 1990 🏛 John Wiley and Sons 🌐 English ⚖ 290 KB

## Abstract We find all possible lengths of circuits in Cayley digraphs of two‐generated abelian groups over the two‐element generating sets and over certain three‐element generating sets.

On Tame Kernel and Class Group in Terms
On Tame Kernel and Class Group in Terms of Quadratic Forms
✍ Qin Yue 📂 Article 📅 2002 🏛 Elsevier Science 🌐 English ⚖ 181 KB

The paper is to investigate the structure of the tame kernel K 2 O F for certain quadratic number fields F ; which extends the scope of Conner and Hurrelbrink (J. Number Theory 88 (2001), 263-282). We determine the 4-rank and the 8-rank of the tame kernel, the Tate kernel, and the 2-part of the clas

Application of Gaschütz' Theorem to rela
Application of Gaschütz' Theorem to relative difference sets in non-abelian groups
✍ John C. Galati 📂 Article 📅 2003 🏛 John Wiley and Sons 🌐 English ⚖ 80 KB

## Abstract Let __G__ be a finite group other than ℤ~4~ and suppose that __G__ contains a semiregular relative difference set (RDS) relative to a central subgroup __U__. We apply Gaschütz' Theorem from finite group theory to show that if __G__/__U__ has cyclic Sylow subgroups for each prime divisor