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I've heard that there is an analogy between algebraic field extensions and covers (in topology). In this analogy Galois extensions correspond to Galois covers and Galois groups correspond to fundamental groups.

Could you explain in more details this analogy and/or give a reference? Moreover, is it just a nice analogy between two different branch of mathematics or are there 'interactions' between them, too?

Thank you in advance.

user72870
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1 Answers1

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Here is a short overview over analogies between Galois Theory and fundamental groups (corresponding objects will be indicated by "$\leftrightarrow$"). Most analogies come from the study of covering spaces. Let $p\colon \tilde{X}\to X$ be a covering.

$X$ $\leftrightarrow$ a field $F$

$\tilde{X}$ $\leftrightarrow$ separated closure $\bar{F}$

$\pi_1(X)$ $\leftrightarrow$ $Gal(\bar{F}/F)$

Theorem 1: There is a bijection between subgroups $H<\pi_1(X,x_0)$ and pointed coverings $(\tilde{X},\tilde{x})\to(X,x_0)$.

Correspondence: bijection between subgroups of $Gal(L/F)$ $\leftrightarrow$ subextensions $L\supset E\supset F$.

Theorem 2: If $N\unlhd \pi_1(X)$, the corresponding covering $p\colon \tilde{X}\to X$ is, s.t. $Homeo_X(\tilde{X})=\pi_1(X)/\pi_1(\tilde{X}),$ where $Homeo_X(\tilde{X})=\mbox{Group of homeos of }\tilde{X}\mbox{ commuting with }p\colon\tilde{X}\to X.$

Correspondence: field extensions, s.t. $L/F$ is Galois and $Gal(L/E)\unlhd Gal(L/F)$. Then $Gal(E/F)=Gal(L/F)/Gal(L/E)$.

Theorem 3: If you consider the monodromy action, you can check, that the cardinality of sheets [which doesn't depend on the chosen point, if the space is path-connected (main argument: compactedness)], equals the cardinality of $\pi_1(X,x_0)/\pi_1(\tilde{X},\tilde{x}_0)$.

Correspondence: $L/F$ Galois extension, $\alpha\in L$, $\mu_\alpha(x)\in F[x]$ its minimal polynom. Let $\{x_i\}$ the set of $\mu_\alpha(x)$. Then $Gal(L/F)$ gives an action on $\{x_i\}$ (analogy of monodromy). There exists also a formula that the cardinality of $\{x_i\}$ equals the cardinality of $Gal(L/F)$ (under certain circumstances) or more general $Gal(L/F)/Gal(L/F[x])$.

Maybe you should search for a theorem which is called "Galois correspondence" (or something like that). I think in Hatcher's "Algebraic Topology"-book (p. 71) you will find something about it.

Daniel Bernoulli
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