Differential Galois theory

간단한 소개

There are two important conditions required in the Galois theory.

transitivity
fixed point free action

Sometimes, \(\text{Gal}(K/F)=|K:F|\) is presented as a condition. This automatically implies fixedpoint free transtive action.

How important is the transitivity in Galois theory and Monodromy theory?

dictionary

geometric viewpoints vs field theoretic viewpoints (or algebra vs. geometry)

covering space - field extensions

regular covering - Galois covering

degree of covering - degree of field extension

Spec of ring of integers - number fields

Riemann surfaces or algebraic curve - function field

Also the permanence of algebraic relation theorem in complex analysis corresponds to the theorem of field theory

If \(\alpha\in\mathbb{\bar{Q}}\) is a root of \(a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots + a_1 x + a_0 = 0, a_i \in \mathbb{Z}\), so is \(\sigma(\alpha)\) where the \(\sigma\) is the automorphism of the algebraic closure.

What about the analytic solution of a differential equation and its continuation?

How to compute the Galois group

we need to decide where it ramifies, i.e. we need information about the discriminanat.
Decide the primes dividing the discriminant.
Look how the primes are interchanged.

regular covering

covering \(p:Y \to X\)

Deck transformation group = \(\pi_1(X)/p_{\sharp}\pi_1(Y)\)

\(\pi_1(X)\) is detecting the information between \(X\) and its universal covering.

If this is regular covering, then the group on the right side is same as the Galois group.

So we can understand the Galois group from homotopic theoretic viewpoint.

the role of algebraic closure

this is analogous to the universal covering.

Suppose we have field extensions \(\bar{F} , K, F\).

We associate some imaginary space \(X_F\) to the field \(F\).

And get a projection \(p:X_K \to X_F\)

Then we define the absolute Galois group \(\text{Gal}(\bar{F}/F)=\pi_1({X_F})\) as a homotopic concept.

Now \(\text{Gal}(K/F)=\pi_1(X_F)/p_{\sharp}\pi_1(X_K)\) is defined.

homotopy lifting theorem

We have a covering map \(p:Y \to X\).

A loop in X can be lifted into Y.

Then it defines an action on Y.

This action is trivial if this loop in X is the image of a loop in Y.

analogy between the circle and the finite field

the field theoretic analogue of the circle = finite field

Galois theory for prime ideals

example

\(x^3-2=0\)

\(K=\mathbb{Q}(\omega, \sqrt[3]{2})\) over \(\mathbb{Q}\)

\(Spec \mathbb{Z}[\omega,\sqrt[3]2]\) over \(Spec \mathbb{Z}\)

\([K : \mathbb{Q}]=6\)

Note that this degree is not equal to the degree of the polynomial.

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표준적인 도서 및 추천도서

Abel_s_theorem_by_Arnold.djvu
arnold book on abel theorem problem 348

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