# Summary. Integral Domains: Remarks and Examples

**Divisor of zero**U is a divisor of zero iff there is

V ≠ 0 such thatUV = 0. We found that neither**Z**norZ[√m] , where m is not a complete square of an integer, have no divisors of zero. But some algebraic structures do.**Z**is a usual notation for the set of residues modulo m. Then, for example, in_{m}**Z**, 2·3 = 0. So, in_{6}**Z**, both 2 and 3 are divisors of zero._{6}**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? Form > 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}= (a^{2}+ b^{2})(c^{2}+ d^{2}),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(A^{2}) = 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}= (a^{2}- mb^{2})(c^{2}- md^{2})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 thatUV = 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

**Z**, 2 and 3 are divisors of zero and, therefore, are not invertible. (For if_{6}UV = 1 butUW = 0 withV ≠ 0, thenUVW = 0·V = 0 , or1·W = W = 0. Contradiction.) On the other hand,5·5 = 1 (mod 6). Therefore, in**Z**, 5 is a unity. The remaining non-zero element 4 is a divisor of zero. For_{6}4·3 = 12 = 0 (mod 6). Also,4·4 = 4 (mod 6). Such elements are called*idempotent*s.*By induction*, in**Z**,_{6}4 This is true for any positive n.^{n}= 4.We may try to consider extensions of finite rings, like

**Z**._{m}Z because, e.g.,_{7}[√2] =**Z**_{7}3 On the other hand, if, as usual, i is taken to be a square root of -1, then^{2}= 2 (mod 7).Z because_{5}[i] =**Z**_{5}2 ^{2}= 4 = -1 (mod 5).

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

- Integral Domains: Strange Integers
- Strange Integers, divisors and primes
- Integral Domains: Fundamental Theorem of Arithmetic
- Integral Domains, Gaussian Integer, Unique Factorization
- Summary. Integral Domains: Remarks and Examples

- Reduction: Constructible Numbers

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