Finite: ℤ/m and the places

Hypercomplex algebras introduces i_k, j_k and ε_k as formal generators with a square attached. But you do not have to adjoin them — ℤ/m already contains them, and which ones it contains is decided one prime at a time. This page follows that all the way down.

A square root of 1 is a root in every channel

The Chinese Remainder Theorem splits the ring: for ,

and a ring isomorphism carries to the same equation in each factor independently. So a square root of 1 mod m is a choice of root in each channel — and an odd prime power has exactly two, , because is cyclic and so has a unique element of order 2.

Counting is then immediate. For odd m,

with the number of distinct primes dividing m. These residues are exactly the sign vectors over the CRT channels — a spectral basis, one coordinate per place. Two of them, and , are the trivial pair; every other one is a genuine split unit, a square root of 1 that is not .

#Sign vectors, as residues

15 = 3·5 gives 2² = 4; 1155 = 3·5·7·11 gives 2⁴ = 16 — one residue per ± choice across four places.

source

<notatio-cell value="PowerModList(1, 1 / 2, 15)" />
<notatio-cell value="PowerModList(1, 1 / 2, 1155)" />

The rows of that second list are the sign vectors: 1 is , 1154 is , and the fourteen in between are the mixed ones. Reading a residue as a vector of signs is the whole content of the CRT here.

The 2-adic channel is the exception

The clean law is stated for odd m because 2 misbehaves, and it misbehaves in a way worth seeing: has one root of 1, has two, and for has four — namely and . The unit group stops being cyclic at 8, so it acquires a second independent element of order 2.

#Powers of two break the count

source

<notatio-cell value="PowerModList(1, 1 / 2, 2)" />
<notatio-cell value="PowerModList(1, 1 / 2, 4)" />
<notatio-cell value="PowerModList(1, 1 / 2, 8)" />
<notatio-cell value="PowerModList(1, 1 / 2, 16)" />

Imaginary units are rarer, and local

Now ask the same question for . Again it decomposes channel by channel, but now most channels have nothing to offer: has a square root of iff. So

Half the primes qualify, by Dirichlet. This is the asymmetry that makes split units the generic case and imaginary units special — and it is the same asymmetry that makes split while does not.

#Where √−1 lives

65 = 5·13, both ≡ 1 (mod 4), so two channels each contributing two roots. 15 = 3·5 has none: 3 ≡ 3 (mod 4) kills it.

source

<notatio-cell value="PowerModList(-1, 1 / 2, 5)" />
<notatio-cell value="PowerModList(-1, 1 / 2, 65)" />
<notatio-cell value="PowerModList(-1, 1 / 2, 15)" />

Hensel lifting up a prime power

Finding a root mod is a search; getting it mod is Hensel's lemma. If then the correction

is a root one power higher, and is invertible because is odd and . Each step is Newton's method on over the p-adics, and it converges for exactly the reason Newton's method does — which is why a root mod is enough to give a root mod for every , and in the limit a root in .

Both of these are one head: Wolfram's PowerModList(a, 1/r, m), the -th roots of mod . The split units are its case and the imaginary units its case, so there is no bespoke head to learn — and the same call reaches past squares.

#One head, any root

Three cube roots of 8 mod 13, because 3 divides 12.

source

<notatio-cell value="PowerModList(8, 1 / 3, 13)" />

#2² ≡ −1 (mod 5), lifted

7² = 49 = 50 − 1, so 7 is the lift of 2 to ℤ/25; 182 is the lift to ℤ/625.

source

<notatio-cell value="PowerModList(-1, 1 / 2, 5)" />
<notatio-cell value="PowerModList(-1, 1 / 2, 25)" />
<notatio-cell value="PowerModList(-1, 1 / 2, 625)" />

The identities transport

A split unit is not just a curiosity of ℤ/m: it is a homomorphism. If in ℤ/m then

respects addition and multiplication, so every identity the symbolic algebra proves becomes a fact about ℤ/m. The zero divisor and the idempotent both survive the trip: becomes , and — an idempotent because — lands on , with .

#The same statement, twice

Left: the symbolic identity. Right: its image under j ↦ 4 in ℤ/15.

source

<notatio-cell value="(1 + j_1) * (1 - j_1)" />
<notatio-cell value="(1 + 4) * (1 - 4) % 15" />
<notatio-cell value="((1 + j_1) / 2) ^ 2" />
<notatio-cell value="10 * 10 % 15" />

This is why the idempotents matter. The split units, the CRT sign vectors and the orthogonal idempotents are three descriptions of one thing: a decomposition of the ring into independent channels.

Split, inert, degenerate — the square as a local invariant

Step back and the three columns of the grid stop looking like three arbitrary choices. Over a field, the quadratic algebra has discriminant , and exactly three things can happen:

	algebra	unit
a square in	— splits	j
a non-square	a quadratic field	i
zero	— degenerate, the two roots collide	\epsilon

So j, i and ε are not three unrelated inventions: they are the three outcomes of one construction, selected by whether is a square, a non-square, or zero. And "is a square" is a local question — it has a different answer at each place.

That is what the mod-m results above are really saying. Take :

• at the real place, is not a square, so — inert;
• at , is a square in , so the algebra splits;
• at it is not, and the algebra is the unramified quadratic extension of ;
• at it is the awkward case, as it was in the counting above.

Whereas splits at every place, which is why split units are everywhere and already at the real place. A hypercomplex unit's character is not a property of the unit — it is a property of the unit at a place.

Where this is heading: profinite and adelic

The rings are not separate objects. They form a system under divisibility — whenever — and the inverse limit of that system is the profinite integers

so the CRT splitting at each finite level is the shadow of one product over all primes. Under that identification the split units assemble too: the elements of order dividing 2 in are a sign per prime, — a Cantor-set of split units, of which the residues above are the finite truncations. Hensel lifting is the statement that a root mod already determines its coherent lift into , so "which units exist" stabilises per place.

Adjoining the archimedean place gives the adeles, , and the split/inert/degenerate trichotomy above is then a statement about all places at once: a quadratic algebra over has a local character at each place, and those characters are not independent — quadratic reciprocity is precisely a constraint tying them together, so the local behaviours have to multiply out consistently. Which means the ± sign vectors that index the split units of ℤ/m are a fragment of something global, not just bookkeeping per modulus.

That is the direction, stated honestly as a direction: this page demonstrates the finite levels and the per-place criterion, and nothing here computes an adelic object. The pieces that would make it concrete — a Norm that names its ambient level, coefficients that live in ℤ/m rather than only mapping into it, and a per-place signature rather than one global square — are all still to build.