Based on what I understand, they are a generalization of prime ideals, and also are related to absolute values. How so?
In: 4
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.
For every prime number, you can make an “absolute value” that is unique to that number. [Ostrowski’s Theorem](https://en.wikipedia.org/wiki/Ostrowski%27s_theorem) says that any absolute value is either the ordinary one *or* these prime-absolute values. If there were *only* prime-absolute values then we could just say that absolute values are primes. But there’s one more, the regular absolute value, so we can’t. We call them “places” instead.
We use the term “place” because we can construct similar kinds of absolute values on algebraic curves. There we actually see a similar patter, where all-but-one of these absolute values correspond to something nice and easy, in this case: Points on the curve. This extra point actually corresponds to the “point at infinity” for the curve. Interestingly, we can simply look at the curve differently to change where the “point at infinity” is and make a point at infinity look like any other point and, therefore, the corresponding absolute value shares the same properties. That is, the extra absolute value only looks differently because of how we decided to write down the curve.
Back in the case of numbers, it’s not so simple. The extra “place” *is* fundamentally different and we just can’t write numbers down differently to fix it. But all of this does suggest that the regular absolute value “should” correspond to some “prime at infinity”. There are actually many different frameworks out there to try and make sense of the notion of a “prime at infinity” which we only have access to because we have absolute values, or places. Ideally, we’d want a frame work where ALL of these absolute values pop out as special cases of a more general idea, but we haven’t yet figured out how.
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