Eisenstein’s Proof Using Complex Analysis

Oswald Baumgart

Chapter Chapter 4 in The Quadratic Reciprocity Law, 2015, pp 41-44 from Springer

Abstract: Abstract 1. Let p and q be two distinct positive odd primes and r the positive minimal residues modulo q; then we will have pr ≡ r ′ or $$\mathit{pr} \equiv -r^{{\prime}}\bmod q$$ , where r ′ again denotes the positive minimal residues modulo q.1 Thus we have $$\displaystyle{\frac{\mathit{pr}} {q} = \frac{r^{{\prime}}} {q} + f\quad \text{or}\quad = -\frac{r^{{\prime}}} {q} + f,}$$ where f and f ′ are integers. This implies $$\displaystyle{\sin \Big(p\frac{2r\pi } {q} \Big) =\sin \frac{2r^{{\prime}}\pi } {q} \quad \text{or}\quad -\sin \frac{2r^{{\prime}}\pi } {q}.}$$ The properties of the sine function expressed by the above equation immediately leads to the result $$\displaystyle{\mathit{pr} \equiv \frac{\sin \frac{2r^{{\prime}}p\pi } {q} } {\sin \frac{2r^{{\prime}}\pi } {q} } \bmod q,}$$ and this in turn shows that $$\displaystyle{p^{\frac{q-1} {2} }\prod r \equiv \prod r^{{\prime}}\prod \frac{\sin \frac{2r^{{\prime}}p\pi } {q} } {\sin \frac{2r^{{\prime}}\pi } {q} } \bmod q,}$$

Keywords: Number Theory; Simple Calculation; Complex Analysis; Sine Function; Minimal Residue (search for similar items in EconPapers)
Date: 2015
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-16283-6_4

Ordering information: This item can be ordered from
http://www.springer.com/9783319162836

DOI: 10.1007/978-3-319-16283-6_4

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().