Based on what I understand, they are a generalization of prime ideals, and also are related to absolute values. How so?
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This is a bit deep for ELI5. so please excuse not fully going there:
You may wonder what functions behave exactly like the absolute value we all know:
– |x| is always a real number and never negative.
– |x| = 0 if and only if x=0 (**positivity**).
– |x+y| ≤ |x| + |y| (**triangle inequality**).
– |x·y| = |x| · |y| (**multiplicative**).
We call such a function a **(generalized) absolute value**. An example is the **standard absolute value** on rational, real or complex numbers. But there are more, for example there is the **trivial absolute value** |-|_0:
|x|_0 = 1 except if x = 0, where positivity forces us to set it to 0.
One can note a bunch of properties. For example, |1| = 1 = |-1| is always true, and multiplicativity also gives us |m/n| = |m| / |n|.
It turns out that if |-| satisfies the above, then so do their s-th power (that is: send x to |x|^^s), at least if s ≥ 1. We then call two absolute values _equivalent_ if each is a power of the other. This becomes relevant if we look at integers or rational numbers:
**Ostrowski’s Theorem**: up to equivalence, the absolute values on ℤ and ℚ are:
– the trivial absolute value |-|_0,
– the standard absolute value |-|_∞,
– for each prime number p the **p-adic absolute value** defined as follows:
If m is any integer, let v(m) count how often m is divisible by the prime p. For example with p=3, we have v(1) = 0 = v(2), v(3) = 1 = v(-3), v(54) = 3. It stands to reason that 0 can be divided by p as often as we want, prompting us to set v(0) = ∞.
Then the _p-adic absolute value_ of m is |m|_p = p^-v(m); note that p^-∞ = 0 returns the correct value |0| = 0. One now checks that this indeed satisfies all four rules. It extends to rational numbers m/n by the aforementioned |m/n| = |m| / |n|.
So, what Ostrowski’s Theorem shows is that absolute values effectively recover prime numbers “p”, but add two more “primes” we denote by “0” and “∞”. This finally leads to the following definition:
A **place** on a set of numbers is a _nontrivial_ absolute value, up to equivalence.
As we saw, the places on ℤ and ℚ are the standard prime numbers and a ominous ∞ one. Often the value is already predetermined by the rules and other values. For example, if we already established |p| = 1/p as in the p-adic one, then it follows from multiplicativity that |√p| = 1/√p = p^-½; in sloppy terms: √p is divisible by p exactly ½ times!
It turns out that in many cases such as _number fields_, the places correspond to _prime ideals_ (called the **finite places**) by counting divisibility as before, and a finite number of **infinite places**. Hence why they somewhat generalize prime ideals.
To see an example of two different infinite places, take the numbers of type a+b√2, with a and b integers or rational. Then taking the “obvious” absolute value |a+b√2| satisfies what we want. But so does the “conjugate” absolute value |a+b√2|’ = |a-b√2|. They are not always the same, for example |1+√2| = 2.41… while |1-√2| = 0.41…
Places satisfy a lot of interesting properties. For example if one _normalizes_† the places, they satisfy the
**Product rule**: for x ≠ 0, the product of all the values |x|_v, with v running over all places including the infinite ones, is always 1!
This somewhat corresponds to prime factorization. For example, x=-12 has standard (“infinite”) absolute value 12, 2-adic absolute value 2^^-2 = 1/4 and 3-adic absolute value 3^^-1 = 1/3. All other ones are already 1. Multiplying them gives indeed 12 · 1/4 · 1/3 = 1.
†: Going into details would be a bit technical, but in most cases, it suffices to require that on the rationals, they are exactly one of the above ones, **not** any power of them.
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