Summary. Integral Domains: Remarks and Examples

Divisor of zero
U is a divisor of zero iff there is V ≠ 0 such that UV = 0. We found that neither Z nor Z[√m], where m is not a complete square of an integer, have no divisors of zero. But some algebraic structures do. Z_m is a usual notation for the set of residues modulo m. Then, for example, in Z₆, 2·3 = 0. So, in Z₆, both 2 and 3 are divisors of zero.
Function N
In the theory of numbers function N is called a norm. It is hard to overestimate the importance of this function. Function N has been used in virtually every single proof so far. Why was it needed? For m > 0, Z[√m] is dense on the real line. The proof uses the Pigeonhole principle in a manner similar to that of a lemma we proved when shredding the torus. This is the reason that we had to use the norm in Euclid's algorithm. Had we used the magnitude of a number, the algorithm would have never stopped.

For m = -1, N(AB) = N(A)N(B), when written explicitly, takes the form

(ac - bd)² + (ad + bc)² = (a² + b²)(c² + d²),

A = a + ib, and B = c + id. This is the two-squares identity. It asserts an interesting fact: a product of the sum of two squares is itself the sum of two squares. If A = B, then N(A²) = N(A)². On the left we have the sum of two squares. On the right - a square of an integer. This is the way to generate Pythagorean triples.

For m positive, N(AB) = N(A)N(B) also has a nice form:

(ac + bd)² - m(ad + bc)² = (a² - mb²)(c² - md²)

This is similar to the two-squares identity. The product of two numbers in the form "a square minus a square time an integer" is an integer in the same form.
Unity
A unity (or just a unit) is a number that divides all other numbers. This is equivalent to saying that it divides 1. If U is a unity, there exists V such that UV = 1. So that U has an inverse. The opposite is also true. Every invertible element is a unity. In a field, all elements, besides 0, are invertible. In a field division becomes ubiquitous and, therefore, not interesting.

In Z₆, 2 and 3 are divisors of zero and, therefore, are not invertible. (For if UV = 1 but UW = 0 with V ≠ 0, then UVW = 0·V = 0, or 1·W = W = 0. Contradiction.) On the other hand, 5·5 = 1 (mod 6). Therefore, in Z₆, 5 is a unity. The remaining non-zero element 4 is a divisor of zero. For 4·3 = 12 = 0 (mod 6). Also, 4·4 = 4 (mod 6). Such elements are called idempotents. By induction, in Z₆, 4ⁿ = 4. This is true for any positive n.

We may try to consider extensions of finite rings, like Z_m. Z₇[√2] = Z₇ because, e.g., 3² = 2 (mod 7). On the other hand, if, as usual, i is taken to be a square root of -1, then Z₅[i] = Z₅ because 2² = 4 = -1 (mod 5).

Constructible Numbers, Geometric Construction, Gauss' and Galois' Theories

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