I was looking for a reference that illustrates a $\mathbb{Q}$vector space basis for the field of padic numbers under the following action. Given a rational number $q$. write, $q=\frac{m}{n}$ where $n>0$. Then, for $x\in \mathbb{Q_p}$ $qx=y$ where $y\in \mathbb{Q_p}$ is the unique element s.t. $mx=ny$.
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The arguments should mirror those for the reals. For example: If you have a Hamel basis for $\mathbb Q_p$, then you can construct a set that fails the property of Baire, but it is consistent with ZF that every set in a Polish space has the property of Baire. 


I'm not sure what is meant by "illustrating" a basis, but the axiom of choice is needed even to prove the existence of a basis for $\mathbb Q_p$ over $\mathbb Q$. 

