The $2$-adic Absolute Galois Group

Find a presentation for the absolute Galois group of the field of 3-adic numbers. It may be helpful to know that the maximal pro-3 quotient of the absolute Galois group is a free group on 2 generators, and that for every finite tamely ramified extension $K/\mathbb{Q}_3$ the maximal pro-3 quotient of $\Gal(\overline{K}/K)$ is a Demuskin group. You should give your output as two or three sections, separated by blank lines: variables, definitions (optional) and relations. 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 $\tau^\sigma = \tau^3$ (which should not be included in the \texttt{relations} section). All later generators should be contained within the $3$-core of the group (the intersection of all $3$-Sylow subgroups). 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. 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. Expressions in profinite groups may involve exponents lying in $\hat{\Z}$. To specify such a constant, you should provide its reduction modulo $5354228880 = 2^4 \cdot 3^2 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23$. Example format for the solution (this is the presentation for the maximal pro-2 quotient of the 2-adic absolute Galois group z,tau,x,y x^2*y^3*y^z

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. You should give your output as two or three sections, separated by blank lines: variables, definitions (optional) and relations. 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). 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. 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. Expressions in profinite groups may involve exponents lying in $\hat{\Z}$. To specify such a constant, you should provide its reduction modulo $85667662080 = 2^8 \cdot 3^2 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23$. Example format for the solution (this is the presentation for the presentation when p=3) sigma,tau,x0,x1 projp = 12880 proj2 = 1701 x0part = sigma^-1*x0^-1*sigma * (x0*tau*x0^-1*tau)^projp sigma2 = sigma^proj2 tau2 = tau^proj2 stau2 = sigma2*tau2 tau4 = tau2^2 stau4 = sigma2*tau4 tau8 = tau4^2 tau16 = tau8^2 inner = (x1 * tau16)^(2*projp) outer = (inner * stau2^2 * inner^-1 * stau2^2)^projp y1 = x1^tau8 * inner^stau2 * outer^stau4 * outer^tau4 x1part = x1^4 * y1 * x1^-1 * y1^-1 x0part * x1part