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 S 2 + iWPHP(Σ b 1), and use it to derive Fermat’s little theorem and Euler’s criterion for the Legendre symbol in S 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 I∆0 + Count2(∆0) with modulo-2 counting principles.