The Intersection of Galois Theory and Algebraic Geometry: Exploring the Orbit Space Structure of Spec F[T]

The first example

Let $\iota :\mathbb{Q}\hookrightarrow \mathbb{Q}[\sqrt{2}]$$\iota:\mathbb Q\hookrightarrow\mathbb Q[\sqrt 2]$ be a field extension, $\mathrm{Gal}(\mathbb{Q}[\sqrt{2}]/\mathbb{Q})\cong \mathbb{Z}/2\mathbb{Z}$$\mathrm{Gal}(\mathbb Q[\sqrt 2]/\mathbb Q)\cong\mathbb Z/2\mathbb Z$​.

We could induce a ring homomorphism

$$\begin{array}{}\text{(1)}& {\iota }^{\prime }:\mathbb{Q}[T]\hookrightarrow \mathbb{Q}[\sqrt{2}][T]\end{array}$$

and act the spectrum functor both sides

$$\begin{array}{}\text{(2)}& \mathrm{Spec}({\iota }^{\prime }):\mathrm{Spec}(\mathbb{Q}[\sqrt{2}][T])\to \mathrm{Spec}(\mathbb{Q}[T])\end{array}$$

Consider $b\ne 0,(T+(a+b\sqrt{2})),(T-(a-b\sqrt{2})),\in \mathrm{Spec}(\mathbb{Q}[\sqrt{2}][T])$$b\ne 0,(T+(a+b\sqrt 2)),(T-(a-b\sqrt 2)),\in\mathrm{Spec}(\mathbb Q[\sqrt 2][T])$,

$$\begin{array}{}\text{(3)}& \mathrm{Spec}({\iota }^{\prime })((T-(a+b\sqrt{2})))=\mathrm{Spec}({\iota }^{\prime })(T-(a-b\sqrt{2}))=(T^2-2aT-a^2+b^2)\end{array}$$

and we could view the roots of irreducible polynomial $f(x)$$f(x)$ as fiber at ${\mathfrak{p}}_f$$\mathfrak p_f$.

$$\begin{array}{}\text{(4)}& p^{-1}(T^2-2aT-a^2+b^2)=\{(T+(a+b\sqrt{2})),(T-(a-b\sqrt{2})\}\end{array}$$

Let $x\in p^{-1}({\mathfrak{p}}_f)$$x\in p^{-1}(\mathfrak p_f)$, then $p\circ \sigma (x)=p(x)$$p\circ\sigma(x)=p(x)$, hence $\sigma (x)\in p^{-1}({\mathfrak{p}}_f)$$\sigma(x)\in p^{-1}(\mathfrak p_f)$ as well.

This is a geometric interpretation of the fact that the Galois group permutes the roots of $f(x)$$f(x)$.

In this case, $p^{-1}(T^2-2)=\{(T-\sqrt{2}),(T+\sqrt{2})\}$$p^{-1}(T^2-2)=\{(T-\sqrt 2),(T+\sqrt 2)\}$ correspond to the orbit of $\sqrt{2}$$\sqrt 2$ under $\mathrm{Gal}(\mathbb{Q}[\sqrt{2}]/\mathbb{Q})$$\mathrm{Gal}(\mathbb Q[\sqrt 2]/\mathbb Q)$.

Fiber functor.

Consider the slice category $B/\mathrm{Top}$$B/\mathrm{Top}$. Let $s$$s$ be a point in $B$$B$, then we can define a fiber functor ${\mathcal{F}}_s$$\mathcal F_s$ as follows:

$$\begin{array}{}\text{(5)}& {\mathcal{F}}_s({\pi }_X:X\to B)={\pi }_X^{-1}(s)\end{array}$$

For a morphism

$$\begin{array}{}\text{(6)}& f:(X\stackrel{{\pi }_X}{\to }B)\to (Y\stackrel{{\pi }_Y}{\to }B)\end{array}$$

We have

$$\begin{array}{}\text{(7)}& {\mathcal{F}}_s(f)=f_s:{\pi }_X^{-1}(s)\to {\pi }_Y^{-1}(s)\end{array}$$

Splitting field

Let $f\in F[T]$$f\in F[T]$ be an irreducible polynomial and $L$$L$ be the splitting field of $f$$f$, and $f$$f$ is separateble in $L$$L$.

Then we could consider the following diagram:

Again, for a point ${\mathfrak{p}}_h=(h(T))\in \mathrm{Spec}(F[T])$$\mathfrak p_h=(h(T))\in\mathrm{Spec}(F[T])$, the fiber correspond to the roots of $h(T)$$h(T)$ in $L$$L$.

$$\begin{array}{}\text{(8)}& p^{-1}({\mathfrak{p}}_h)=\{(T-\alpha ),(T-{\sigma }_1(\alpha )),...,(T-{\sigma }_n(\alpha ))\}\cong \mathrm{Spec}(L[T]/(h(T)))\cong {\mathcal{O}}_{\mathrm{Gal}(L/K)}(\alpha )\end{array}$$

The fiber functor ${\mathcal{F}}_{(f)}$$\mathcal F_{(f)}$ shows that why the Galois group permute the roots of $f$$f$ via ${\sigma }_f:p^{-1}(f)\to p^{-1}(f)$$\sigma_f:p^{-1}(f)\to p^{-1}(f)$.

Orbit Space

Let $G$$G$ be a group and let $G$$G$ acts on $X$$X$ via $\varphi :G\to {\mathrm{Aut}}_{\mathcal{C}}(X)$$\phi:G\to\mathrm{Aut}_{\mathcal C}(X)$. We denote that $X/G$$X/G$​ to be the space of orbit of this group action.

For all the $G$$G$ invariant function, i.e. $\mathrm{\forall }g\in G,f(g\cdot x)=f(x)$$\forall g\in G,f(g\cdot x)=f(x)$, we have an unique function $\hat{f}:X/G\to X$$\widehat f:X/G\to X$ such that

$$\begin{array}{}\text{(9)}& f=\hat{f}\circ \pi \end{array}$$

Now think about the splitting field of $f\in F[T]$$f\in F[T]$ again.

Consider the orbit space

$$\begin{array}{}\text{(10)}& \mathrm{Spec}(L[T])/\mathrm{Gal}(L/F)\end{array}$$

We know that the Galois group act transitive at the roots of $f$$f$ since $f$$f$ is irreducible.

Then a irreducible polynomial $g$$g$ split in $L$$L$ iff $g=\prod _{\sigma \in \mathrm{Gal}(L/F)}(T-\sigma (\alpha )),\alpha \in L-F$$g=\prod_{\sigma\in \mathrm{Gal}(L/F)}(T-\sigma(\alpha)),\alpha\in L-F$.

Hence the orbit space

$$\begin{array}{}\text{(11)}& \mathrm{Spec}(L[T])/\mathrm{Gal}(L/F)\cong \mathrm{Spec}(F[T])\end{array}$$

So the Galois Group ''glue them back''.

The $p$$p$ can be viewed as the projection from $X$$X$ to $X/G$$X/G$.

$$\begin{array}{}\text{(12)}& p:\mathrm{Spec}(L[T])\to \mathrm{Spec}(L[T])/\mathrm{Gal}(L/F)\end{array}$$

Hence we get the universal property of the affine scheme $\mathrm{Spec}(F[T])$$\mathrm{Spec}(F[T])$: