Your query caused meto check out the offerings at REI. All traditional ice axes were $100 or lessandwere two brands - Grivel and Black Diamond. Both are respected companies who do not sell junk. All should be adequately strong for their intended use.

I have found an ice axe useful for general walking, especially if I needed its capabilities further on during the trip. Just be aware that an ice axe is an edged tool and should be handled with care. A cover for the pick is absolutely necessary.

If you really want to go all out, check out some of the futuristic ice tools. They are more expensive, so obviously they must be more capable (sarcasm alert!). Actually they work for the more extreme types of ice climbing.