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Jacobi symbol one does not imply a square: $(2 / 15) = 1$

Statement refuted

The converse of A unit square modulo an odd integer has Jacobi symbol one is false: a unit $a$ with $(\frac{a}{n}) = 1$ need not be a square modulo the odd positive integer $n$ .

Facts & Assumptions

Given: The unit numerator $2$ and the odd denominator $15 = 3 \cdot 5$ .

[L1]

For odd $n \geq 1$ with canonical prime factorisation $n = \prod_{i < r} p_{i}^{e_{i}}$ , define $(\frac{a}{n}) : = \prod_{i < r} {(\frac{a}{p_{i}})}^{e_{i}}$ (The Jacobi symbol, with its zero value and empty-product convention).

Counterexample

technique · direct

1.1

The nonzero square classes modulo $3$ are ${1}$ and those modulo $5$ are ${1, 4}$ , so $(2 / 3) = - 1$ and $(2 / 5) = - 1$ . Using $15 = 3 \cdot 5$ in [L1] gives $(2 / 15) = (- 1) (- 1) = 1$ .

L1givenalgebra

2.1

If $x^{2} \equiv 2 (m o d 15)$ , reduction modulo $3$ would give $x^{2} \equiv 2 (m o d 3)$ , but the complete square set modulo $3$ is ${0, 1}$ . Thus $2$ is not a square modulo $15$ , despite its Jacobi symbol being $1$ .

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 47 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

P. Hackman, Elementary Number Theory, §D.II (standard reference, not scraped)
A. Gorodnik, Number Theory, Lecture 10, Note 1.3 (standard reference, not scraped)

Jacobi symbol one does not imply a square: (2/15)=1(2/15)=1