TITLE: Parametrizing all cyclic extensions of the rational numbers of degree 5
QUESTION [11 upvotes]: Is there a polynomial $f(T,X) \in \mathbb{Q}(T)[X]$ in the indeterminate $X$ over the field $\mathbb{Q}(T)$ with $\mathrm{Gal}(f/\mathbb{Q}(T)) \cong \mathbb{Z}/5\mathbb{Z}$ such that for every Galois extension $K/\mathbb{Q}$ with $\mathrm{Gal}(K/\mathbb{Q}) \cong \mathbb{Z}/5\mathbb{Z}$ there exists some $t \in \mathbb{Q}$ such that the splitting field of $f(t,X)$ over $\mathbb{Q}$ is isomorphic to $K$.

REPLY [11 votes]: Evidently no.  In "Generic Polynomials Constructive Aspects of the Inverse Galois Problem" by Jensen, Ledet, Yui (2002): generic dimension (i.e. minimum number of parameters in a generic polynomial) of $\mathbb{Z}/5$ over $\mathbb{Q} = 2$ (p.203) and moreover essential dimension (i.e. minimum transcendence degree of parameters) of $\mathbb{Z}/5$ over $\mathbb{Q} = 2$ (p.190).  They conjecture (p.202) that if generic dimension is finite then essential dimension = generic dimension (over any field).  One story for $\mathbb{Z}/3$ that breaks down for $\mathbb{Z}/5$ uses an action of $\mathbb{Z}/3$ on $\mathbb{P}^1_{\mathbb{Q}}$. 
Edit: As Pablo noted, this answer is about a generic polynomial over $\mathbb{Q}$ but it doesn't answer the original question about existence of a 1-parameter parametric polynomial over $\mathbb{Q}$.  I didn't appreciate this nuance in the original question when I wrote the above answer.