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The signed half-system for $3$ modulo $11$ gives $(3 / 11) = 1$

Example

For $p = 11$ and $a = 3$ , the signed half-system is

$3, - 5, - 2, 1, 4.$

There are two negative signs, so Gauss's lemma gives $(3 / 11) = 1$ .

Facts & Assumptions

Given: The odd prime $11$ , multiplier $3$ , and the half-system $1, 2, 3, 4, 5$ .

[L1]

Multiplication by a unit modulo an odd prime reduces the half-system to unique signed representatives whose absolute values permute the half-system (Multiplication by $a$ with $p ∤ a$ permutes an odd prime's signed half-system up to sign).

[L2]

If $N (a, p)$ counts the least positive residues of $a j$ that exceed $p / 2$ for $1 \leq j \leq (p - 1) / 2$ , then $(a / p) = (- 1)^{N (a, p)}$ (Gauss's quadratic-residue lemma).

Verification

technique · direct

1.1L1givenalgebra

The products $3, 6, 9, 12, 15$ reduce modulo $11$ to the signed representatives $3, - 5, - 2, 1, 4$ .

2.1L1step 1.1

Their absolute values are $3, 5, 2, 1, 4$ , a permutation of $1, 2, 3, 4, 5$ as [L1] requires, and exactly two signs are negative.

3.1L2step 2.1∎

Fact [L2] gives $(3 / 11) = (- 1)^{2} = 1$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

H. Hackman, Elementary Number Theory, Chapter D, Section D.IV (standard reference, not scraped)

The signed half-system for 33 modulo 1111 gives (3/11)=1(3/11)=1