is_field

std.algebra.is_field · Level L3

Whether a finite ring is a field, as 1 or 0: a commutative ring with 1 ≠ 0 in which every nonzero element has a two-sided multiplicative inverse, checked for every nonzero element. Division is defined only by nonzero elements. The order of a finite field is a prime power, not necessarily a prime: there is a field with 4 elements, and Z/4 is not one.

commutative, 1 ≠ 0, ∀a ≠ 0 ∃a⁻¹

Signature

is_field(Ad: i64[n, n], Mu: i64[n, n]) → f64[]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference a commutative ring in which every nonzero element is a unit in exact rational arithmetic, on all 40 test cases.
All 40 float64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Identity

Calls

ring_axioms ring_properties

Called by

—

sha256:88aeec34f5ddd2fe4a261d60fa69a4112240f4bc01df792f9478cbfbec6cf2b0

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.algebra.is_field · Ω:is_field
Pins: sha256:e36788352ac456e7a0dc85d9ccb820e8eb9b8247a619c33807e44a6abf55223ethis graph and the 5 it reaches through calls
Evidence: sha256:3925bebc758f5a39da957f975db28c71c1a611b6902f1a3fb800e0c50ecfc5a3the hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.