Alphabeta MathQuadratic Reciprocity and the Jacobi Symbol — Examples
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Group Homomorphisms and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- The Fundamental Theorem of Finite Abelian Groups
Number Theory
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Primitive Roots and Unit Groups Modulo N
- Quadratic Reciprocity and the Jacobi Symbol
- Quadratic Residues and the Legendre Symbol
Real Analysis
2 · Logical flowchart
3 · Items
Example
A complete reciprocity table for 3,5,7,11
Example
Computing (1111/8093) by the Euclidean Jacobi algorithm
Counterexample
Jacobi symbol one does not imply a square: (2/15)=1
Example
Lifting both square roots of 2 from modulo 7 to modulo 7⁴
Example
The four square roots of 9 modulo 128
Example
The square roots of 1 modulo 360 by the Chinese remainder theorem
Counterexample
Odd-prime tests alone miss the obstruction to 13 being a square modulo 24
Example
Odd primes represented by a divisor of x²+3
Corollary
There are infinitely many primes congruent to 1 modulo 3
Sources
- A. Gorodnik, Number Theory, Lecture 10, Example 1.6
- V. Shoup, A Computational Introduction to Number Theory and Algebra, 2nd ed., §12.3
- P. Hackman, Elementary Number Theory, §D.II
- A. Gorodnik, Number Theory, Lecture 10, Note 1.3
- A. Gorodnik, Number Theory, Lecture 9, Example 2.4
- A. Gorodnik, Number Theory, Lecture 9, consequence after Example 2.4