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28 lines
1.9 KiB
Markdown
28 lines
1.9 KiB
Markdown
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Find a presentation for the absolute Galois group of the field of 2-adic numbers. It may be helpful to know that the maximal pro-2 quotient of the absolute Galois group can be presented with three generators x,y,z and one relation, x^2*y^3*y^z=1. The theory of Demuskin groups may also be useful.
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You should give your output as two or three sections, separated by blank lines: variables, definitions (optional) and relations.
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The variables section should consist of a comma separated list of variable names on one line corresponding to the generators of the presentation. The first and second generators (say $\sigma$ and $\tau$ respectively) are distinguished, and must satisfy the implicit relation $\tau^\sigma = \tau^2$ (which should not be included in the \texttt{relations} section). All later generators should be contained within the $2$-core of the group (the intersection of all $2$-Sylow subgroups).
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In the definitions section, you may define additional variables in terms of the generators and variables defined earlier within the definitions section. Each line should take the form "var = expr", where expr is an expression built using group operations.
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In the relations section, you should give the relations as expressions in terms of variables defined in the previous two sections. If there are multiple relations, give one on each line.
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Expressions in profinite groups may involve exponents lying in $\hat{\Z}$. To specify such a constant, you should provide its reduction modulo $85667662080 = 2^8 \cdot 3^2 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23$.
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Example format for the solution (this is a presentation for p=3)
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sigma,tau,x0,x1
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tau2 = tau^3011753745
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sigma2 = sigma^3011753745
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rho = tau2^4
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angle = (x0*tau*x0^-1*tau)^2974571600
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u = (x1*(rho^2)*x1*(rho^2))^2974571600
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v = sigma2*tau2
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w = (u*(v^2)*u^-1*(v^2))^2974571600
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y1 = x1^rho*(u^v)*(w^(sigma2*tau2^2))*(w^(tau2^2))
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comm = x1^-1*y1^-1*x1*y1
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(x0^sigma)^-1*angle*x1^3*comm
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