I have some questions concerning the rationnal numbers and the vector spaces. Let's take the set of rational number Q with the usual addition and multiplication. We can say that (Q,+,.) is a vector space on the Q field. Now, if we add the |x| absolute value, we define have vector space with a norm. If we add to this space all the points of the cauchy convergent sequence, do we have a complete space (i.e. banach space)? If yes, we can call this set Q* (may there is already an other name ;). Is (Q*,+,.) a field? Seratend.