Set $h(x) = x^5+x^4+x^3+x^2+x-1$, let $L$ be the splitting field of $h$ over $\mathbb{Q}$, and let $p$ be a prime of $L$ lying over $2$.



> 
> What is the isomorphism class of the inertia group $I\_p$, and how do I find it?
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> 


I would be more happy with having some procedure for finding inertia groups, rather than only the isomorphism class of this specific group.


I would also like to find the field $O\_L/p$, and the decomposition group $D\_p$ if possible.


Some useful information is: $h$ is irreducible over the rational numbers, and $h(x)(x-1) = x^6 - 2x + 1$ which allows for an easy irreducible factorization mod $2$: $h(x) = (x + 1)(x^2 + x + 1)^2$. 



Finding the inertia group
This answer produces $I$ somewhat indirectly. So it actually *does not* what the OP asked for.


The decomposition group $D$, which is the Galois group of $h(x)$ over $\mathbb Q\_2$, can be computed as follows: Using resultants, one sees that the minimal polynomial over $\mathbb Q$ of the difference of two distinct roots of $h(x)$ is \begin{equation}
H(y)=y^{20}+6y^{18}+21y^{16}+46y^{14}-116y^{12}+694y^{10}+1837y^8-1810y^6-1527y^4+8560y^2+9584.
\end{equation}
From the factorization of $h(x)$ over $\mathbb Q\_2$ we know that $D$ is a transitive subgroup of $S\_4$. Over $\mathbb Q\_2$ the polynomial $H(y)$ factors into irreducibles of degrees $4,4,4,8$. The only transitive subgroup of $S\_4$ which has these orbit lengths on the $20$ pairs of distinct elements of $\{1,2,3,4,5\}$ is the dihedral group of order $8$.


As Will Sawin already remarked, the inertia group $I$ is a subgroup of $S\_2\times S\_2$. On the other hand, $D/I$, as a Galois group of an extension of a finite field, is cyclic. This forces $|I|\ge4$, because a dihedral group of order $8$ modulo a normal subgroup of order $\le2$ isn't cyclic. Thus $I=S\_2\times S\_2$.


There is a [paper](http://dx.doi.org/10.1006/jabr.1994.1067) by Sybilla Beckmann which describes a method to compute inertia groups under certain restrictions. Her theorem does not apply here. The paper finishes with some remarks about how to possibly extend the methods.


