Abelian groups and quadratic residues in weak arithmetic

We investigate the provability of some properties of abelian groups and quadratic residues in variants of bounded arithmetic. Specifically, we show that the structure theorem for finite abelian groups is provable in S2 2 + iWPHP(Σb 1), and use it to derive Fermat’s little theorem and Euler’s criterion for the Legendre symbol in S2 2 + iWPHP(PV ) extended by the pigeonhole principle PHP(PV ). We prove the quadratic reciprocity theorem (including the supplementary laws) in the arithmetic theories T0 2 +Count2(PV ) and I∆0 + Count2(∆0) with modulo-2 counting principles.

• Type: Journal
• Year: 2010
• Where: MLQ
• Authors: Emil Jerábek
• Keywords: Abelian Groups, Fermat’s Little Theorem, MLQ 2010, finite Abelian Groups