How can I eradicate this alien ground cover? It has taken over my whole yard--40 ft. by 80 ft.

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Lesser Celandine

With infestation that big, you've got a big job ahead of you that will no doubt involve both removing it all then re-seeding or sodding your lawn.

Hi, Gregg,
Thank you for asking this particular question. Thanks, rickj, for the link. You have also helped me with a similar problem; with VA Wild Violets! Looks similar, with the tube runners below ground, and the "knobs" attached for each individual cluster of above ground leaves.
This stuff is awesome, growing very, very quickly in the spring and summer.
It is hard to get rid of, but can be done. I have used about every 7 days, a spray consisting of Ortho Max Weed Killer, mixed with a little Dawn Dishwashing Detergent. The detergent helps it stay on the leaves. It will eventually kill it, even the root "Knobs", but it does take about a month before it's gone!
This Max week killer does not kill grass, only weeds.
I agree that the best way would be to possible use a tiller, then using a rake to remove all traces (or as much as possible) of the roots, then go from there.
PS; As an edit, I will look for the Glyphosfate (spelling?) cause I have some stuff behind my back privacy fence that I want to kill... all of it!

"Round Up" (glyphosate) will work in just a few days but will kill everything it touches down to the root. May be worth trying with a big patch like yours, then wait a couple of weeks and seed the area or sod it. Don't spray on a windy day and do not get this product on anything you don't want to kill. Available at Lowe's, Home Depot, and WalMart.