Mice ate my beetroot!

This is our first year on the allotment, i 'had' a lovely row of beetroot that i was looking forward to picking this week. I had noticed damage to my broad beans but wondered about bird damage ( not unexpected ). Now when i pulled my beetroot tonight all of the roots had been eaten! definatly tiny teeth marks on what was left. As you can understand we are absaloutly gutted. Does anyone have any suggestions on how i can keep my veg safe. Im not against traps but id like to try other methods first. What has worked for you?

I had terrible trouble this year with mice they ate all my beetroot and my broad beans 3 times, I am now finding teeth marks in my tomatoes that are in the greenhouse, I did use traps in the end, but they say to try prickly plants around the base of the ones you don't want eaten, try it I did and it helped a little but not enough.  I don't know how they are getting in my greenhouse as there are no holes so I am baffled by this one. I also don't know if you can get mice scarers you seem to be able to get them for most animals, I have not checked this root but may have to before next season.