Ivy in Trees
The solution here is to cut the ivy at the base of the plant as you thought. The ivy in the tree will die off and you should be able to pull it off the trunk.
New ivy growth can be attractive so it may be an idea to simply keep chopping the ivy down to ground level on a regular basis. This will weaken the plant, but will not kill it.
If you really hate it and would like to eradicate it completely, ideally you need to dig out the ivy plant from the ground to prevent it regrowing. However this may prove difficult if it is tangled in the tree's roots, or close to the base of the tree. There are chemicals available, such as Glyphosate, that can be applied to the stump of the ivy that will kill it off, but may need more than one application.
Either way it sounds like a good winter task to get your teeth into in this cold weather!