Prime numbers in complex domains are actually quite simple

Likewise, a number in Z[√−2] is said to be divisible by another number in Z[√−2] if the result of the division is also a number in Z[√−2].This means that numbers that are prime in Z are not necessarily prime in Z[√2], Z[√−2] or other domains of algebraic integers.In particular, given x being a solution to x² + bx + c = 0 with c a prime in Z, it turns out that c is divisible by x in the relevant quadratic integer ring.For example, in Z[√2], we see that 2 is divisible by √2, since 2 divided by √2 is √2..Similarly, in Z[√−2], we see that 2 is divisible by √−2, since 2 divided by √−2 is −√−2.So in both domains, 2 is neither irreducible nor prime..Both √2 and √−2 are prime in their respective domains, as are numbers like 1 + √−2 and 3 − √2 (the former corresponds to x² − 2x + 3 and the integer 3, the latter to x² − 6x + 7 and the integer 7).Both Z[√2] and Z[√−2] are unique factorization domains, and also principal ideal domains..The concept of “ideals” is also a simple concept that would take me more than a half hour to explain.Suffice it to say here that it means that in these two domains all prime numbers are irreducible and all irreducible numbers are also prime..Just like in Z.But in the vast majority of imaginary quadratic integer rings, several numbers have more than one distinct factorization..In fact, Z[√−2] is in a distinct minority, one of only nine rings.As for Z[√2] among the purely real rings, that’s an open question I won’t get into here.The most famous example of multiple distinct factorizations in an imaginary quadratic integer ring is 6 in Z[√−5], where we have (1 − √−5)(1 + √−5) = 6..But 2 and 3 are also irreducible in Z[√−5], and it is also true that 2 × 3 = 6.We verify that 1 + √−5 is not divisible by either 2 or 3 in this ring: the minimal polynomial for 1/2 + (√−5)/2 is 2x² − 2x + 3, and the minimal polynomial for 1/2 + (√−5)/3 is 3x² − 2x + 2.This is just as 6 is not divisible by 4 in Z..But in Z, both 4 and 6 are divisible by 2, and 6 is divisible by 3.Drawing the diagramsFrom high school you probably remember the real number line..It is almost always horizontal, often with 0 at the center, with the negative numbers to the left of 0 and the positive numbers to the right.Generally the real number line is shown with 0 at the center..We might see on the line, for example, −3, −2, −1, 0, 1, 2, 3..The imaginary number line can also be shown with 0 at the center..For example, −3i, −2i, −i, 0, i, 2i, 3i.Since 0 is both purely real and purely imaginary, it only makes sense to make it the juncture of the real and imaginary number lines.. More details

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