In reply to sethmford:
The answer is of course, "not too long and not too short".
Too long and it becomes a pain to use when you really to use it to actually climb, over an ice bulge, crevasse, etc. Too long also becomes bad for ice axe braking. (Long is also heavy)
Too short and it is useless as a walking stick, and again too short is bad for ice axe braking.
I always found from 50 to 55 cm ideal and I'm about your height. I have an old Charlet Moser that I think is actually 53 cm and the perfect length for me. Held in the braking position it ends at exactly the right position for the left hand at the side of your body (with the ferule not under your chest and not poking out too far beyond your chest). Your mileage may differ.
The other question you might be asking is whether to buy a total bumbly walking axe, or something with a more technical pick. While a bumbly axe is cheap it also rapidly becomes useless if you decide to do slighly more technical routes (not icefalls, obviously). So I would say buy something that you can actually climb with (like Scottish grade II and III). That way you have a tool that you can use for the rest of your climbing career (because everbody enjoys occasional easy snow and ice routes even if they become superheroes).