"frost/freeze was supposed to kill the stuff..."

I think you were misinformed about that. Alfalfa is a perennial.

I can't tell you how to kill it other than repeated tilling or plowing or by using an herbicide such as Roundup.

Jim

If you can, take your lawn mower to it and then turn it under but the roots are supposed to go deep on alfalfa so try to get all the roots up and exposed to the sun to dry out.

so I guess the University of Minnesota researchers really didn't find an annual alfalfa superior to the perennial alfalfas

sheesh. 'nother weed in the garden...

Here is a link that might be useful: Big N Alfalfa

I think I would try smothering it. There are several different ways to do this. One is to cover it all with black plastic and wait until it dies. Another is to cover it with wet corrugated cardboard and then cover the cardboard with layers of organic materials. Plant shallow rooted spring crops into the lasagna layers and by summer it will be ready for deeper rooted crops. Potatoes could also be planted on top (see the link below.)

Here is a link that might be useful: Planting potatoes over turf