I'm studying methods of computation of Galois group of irreducible polynomials over $\mathbb{Q}$. In case of fifth degree there are 5 variants of Galois group:$S\_5,A\_5,AGL\_1(\mathbb{F}\_5).D\_5,\mathbb{Z}\_5$. I have troubles with determining wether Galois group is $\mathbb{Z}\_5$ or $D\_5$ if given that it is a subgroup o $D\_5$ because my algorithm requires too big computational power and I wonder if there are an easy way to solve this particular problem. 



Computation of Galois group
Section 6.3 of Henri Cohen's "A Course in Computational Algebraic Number Theory" is about computing Galois groups using resolvents. Subsection 6.3.4 is specifically about the quintic case. Cohen gives an algorithm which addresses the case of $\mathbb{Z}\_{5}$ versus $D\_{5}$ in step 6 (once a 5-cycle contained in the Galois group is known). This algorithm is implemented in PARI/GP (see [this page](http://pari.math.u-bordeaux.fr/)).


In light of David Speyer's remarks, the difficulty is in proving the Galois group is $\mathbb{Z}\_{5}$ when it appears that the only possibilities for the factorization of $f(x)$ modulo $p$ are irreducible, and a product of $5$ linear factors. 


