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Some number theory basics part 2: Another (not so mini) thread
1/n https://twitter.com/anton_hilado/status/1346462114446270464
1/n https://twitter.com/anton_hilado/status/1346462114446270464
Another important concept in number theory is Galois groups. Before we tackle that, let's learn some abstract algebra and set some notation.
The integers, denoted Z, is a "ring". We can add, subtract, and multiply two integers to give another.
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The integers, denoted Z, is a "ring". We can add, subtract, and multiply two integers to give another.
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We can't always divide integers to give another. 4/2 is an integer, but 1/2 is not. The question of which divides which makes integers hard but interesting.
The rational numbers, denoted Q, are a "field". They are a ring but we can also divide (except by 0).
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The rational numbers, denoted Q, are a "field". They are a ring but we can also divide (except by 0).
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By "divide" I mean dividing any rational number by another (except 0) gives a third. This makes them maybe slightly less interesting, but easier!
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The integers Z are the "ring of integers" of Q, while Q is the "field of fractions" of Z. We won't define these terms here but instead just appeal to intuition. The Gaussian integers we defined last time we will denote by Z[i].
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The "Gaussian rationals" are the field of fractions of Z[i]. They are the complex numbers a+bi where a and b are both rational numbers. We denote them by Q(i).
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Now, the Galois group. It is a "group", another concept in abstract algebra, which I won't define but we think of it as the permutations of a big field that don't change a smaller field contained in the big field.
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For instance, all the Gaussian rationals Q(i) contain all of the ordinary rational numbers Q. If we do complex conjugation, this jumbles around the Gaussian rationals, except the ordinary rationals, which stay fixed!
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So, i gets sent to -i. 1+2i gets sent to 1-2i. But something like, say 10 stays fixed! This is one element of the Galois group of Q(i) over Q (the shortcut notation is Gal(Q(i)/Q). Its only other element is the "do nothing" permutation, aka the identity.
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Numbers are hard to understand, so we usually study their "symmetries" instead via Galois groups. These important mathematical objects are named after Évariste Galois, a mathematician from 19th century France with a fascinating but unfortunately short life story.
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Consider two fields, a big field "E" and a smaller field "F" inside of E (in our example E could be Q(i) and F could be Q). We look at Gal(E/F). Consider a prime P in E. The elements of Gal(E/F) that leave it fixed is called the decomposition group (of P).
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Now you can use modular arithmetic to construct "residue fields" which in the case of numbers are finite. This is an entire fascinating subject in itself which I won't be able to cover here. Take the prime p of F which the prime P of E lies over.
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Let O_E and O_F be the rings of integers of E and F respectively. We have the residue fields K_P=O_E/P and k_p=O_F/p (technically we have to make use of the language of ideals here). There's a Galois group Gal(K_P/k_p).
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Wait wait wait why are we considering these "residue fields" and their Galois group?
Because they have an easier structure! They are "generated by one element" which has a special name - "Frobenius". And this easier Galois group we can relate to the harder original one.
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Because they have an easier structure! They are "generated by one element" which has a special name - "Frobenius". And this easier Galois group we can relate to the harder original one.
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There is a map (a function, more properly a "group homomorphism") from the decomposition subgroup of P (a subgroup of Gal(E/F)) to Gal(K_P/k_p). The decomposition subgroup can be more complicated than Gal(K_P/k_p). Mapping into the latter loses some information.
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In particular, a "subgroup" of the decomposition group gets sent to the identity. This subgroup is called the "inertia group". Its presence (if non-trivial) signals the presence of *ramification* (remember this concept from last time?)
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Denoting the decomposition group of P by D_P and its inertia subgroup by I_P, we have an "exact sequence"
1->I_P->D_P->Gal(k_P/k_p)->1
It is a very important part of number theory.
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1->I_P->D_P->Gal(k_P/k_p)->1
It is a very important part of number theory.
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Let's see what happens for the Gaussian integers! Our Galois group has only two elements, so it's easier. Let's pick a prime, say 3. Recall that this remains prime in Z[i]. The residue field K_3 is Z[i]/(3), with 9 elements and k_3 is Z/(3) with 3 elements.
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Let's look at the groups. 3 is fixed by conjugation, so D_3 has 2 elements. Gal(K_3/k_3) has 2 elements, this requires abstract algebra. I_3 has no choice but to be trivial. No ramification.
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What about 5? 5 is not prime in Z[i], so we choose instead one of the ones it splits in, say 1+2i. The residue fields are K_1+2i=Z[i]/(1+2i) with 5 elements and k_5 with 5 elements. 1+2i is fixed by conjugation, so D_1+2i is trivial. Gal(K_1+2i/k_5) is also trivial.
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This actually forces the inertia subgroup to also be trivial. There is no ramification.
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Finally let's look at the prime 1+i, which lies above 2. The residue fields are Z[i]/(1+i) with 2 elements and Z/2 with 2 elements. Now 1+i, the element, gets sent by conjugation to 1-i...but this is itself times a unit! In the language of ideals, this ideal is fixed.
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So what this means is that D_1+i is fixed by the entire Galois group, and has 2 elements. Meanwhile Gal(K_1+i/k_2) is trivial. The inertia I_1+i will have two elements, indicating the presence of ramification! This agrees with what we know.
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So the splitting and ramification behavior of these numbers (maybe we should start calling them by their name "number fields" now) can be studied using Galois groups. We stuck to a very simple case with the Gaussian integers but it can get complicated (and interesting)!
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And that's it for this thread! I oversimplified quite a bit here, but I hope I did the main ideas some justice. Hopefully more number theory to come in the future!
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