Hilbert Symbol Calculator

Restricted realplaceHilbertsymbol(a, b)_infinity,−1iffbothnegative; NOT finitep-adicsymbol. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.

Description

At the real place, a Hilbert symbol depends only on whether both nonzero arguments are negative.

At the real place, a Hilbert symbol depends only on whether both nonzero arguments are negative.^[1]

This tool evaluates that restricted sign rule without computing a finite-prime p-adic symbol.

When to use Hilbert Symbol Calculator

This tool evaluates that restricted sign rule without computing a finite-prime p-adic symbol.

How to use Hilbert Symbol Calculator and how it works

Prepare first real nonzero, second real nonzero, using the input definitions and matching units below.
Enter the required fields and run Hilbert Symbol Calculator.
Read result.realPlaceHilbertSymbol using the units, signs and ordering explained below.
Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.

Input field	Meaning	Value type	Requirement
actualFirstRealNonzero	First finite nonzero argument of the real-place symbol.	number	Required; finite; other constraints below
actualSecondRealNonzero	Second finite nonzero argument of the real-place symbol.	number	Required; finite; other constraints below

Require two finite nonzero real arguments. Return -1 when both are negative and +1 otherwise. Zero is an input error and no finite-prime p-adic Hilbert-symbol branch is implemented.

realHilbertSymbol=(a<0 && b<0)?-1:1

Apply the input ordering, units and branch conditions stated above.

Output field	Meaning	Value type
result	Object containing the calculated or selected fields below.	object
result.realPlaceHilbertSymbol	Integer -1 for two negative arguments, otherwise +1.	number

json

{
  "input": {
    "actualFirstRealNonzero": 1,
    "actualSecondRealNonzero": -2
  },
  "output": {
    "result": {
      "realPlaceHilbertSymbol": 1
    }
  }
}

Limitations and assumptions

The calculation uses the supplied observations and model inputs. It does not collect measurements, select a reference population or coefficient, or establish that the model applies to the actual case. The stated branch conditions and output units remain part of the result.

All fields require explicit values. Unexpected fields, wrong types, nonfinite numbers and sparse arrays are rejected. Alignment, bounds and supported-token rules are explained above; extremely large inputs or nonfinite results can also fail. Input errors differ from successful false comparisons, unavailable-domain flags and empty match lists.

Alternative or Complementary methods

Use Frobenius Symbol Calculator for a complementary operation: A Frobenius symbol requires a specified number-field extension and an unramified prime-ideal context.

Use Wieferich Pair Checker for a complementary operation: A mutual Wieferich pair requires two distinct primes satisfying both directions of a modular-power condition.

References

Hilbert Symbol Calculator — tool.io

Similar or alternative tools

Frobenius Symbol Calculator
A Frobenius symbol requires a specified number-field extension and an unramified prime-ideal context.
Wieferich Pair Checker
A mutual Wieferich pair requires two distinct primes satisfying both directions of a modular-power condition.