first off, if you have room to put it in the ground, by all means do it. you won't have to worry about how to overwinter it in a pot. while in the ground, the surrounding soil will help moderate any extreme temperature swings that may occur throughout the dormant season. all you have to do is to throw an inch or two of some kind of mulch/compost over the crown before winter and you'll be good to go. i've never tried to keep hops in pots (above the soil level) throughout the winter, but i would figure that any thing you could do to keep the crown from experiencing big swings in temperature would be a big plus (a lot of insulation). as for what you snapped off and left in the ground, if it was a true root, it'll just die off and decompose. if it was a rhizome, you'll probably see tiny shoots coming up next spring. here is a good overview or the plant: http://www.oregonhops.org/culture2.html
. good luck and i'm sure you'll be successful.