Yes. As someone growing and raising between 60 and 100% (depending on year) of my family's food on a run-through clay and hardpan soil (150 years of cotton and tobacco did it), I can definitely say that you can turn compacted and dead clay around. Like the parent said - deep mulch is one way. In my case, leaves, grass clippings, forest duff, soiled animal bedding, etc. Really anything organic. You effectively want to "build" the topsoil, which will over time (4-7 years depending on what you're starting with) create enough organic matter to entice insects, earthworms, fungal growth (critical to soil health), voles and moles. All of these creatures will aerate the soil for you. No work required. If you want to do a test, throw a tarp down (or some other large covering) over an area, and check back on it in a week or two. You should have some moles already going to work on aerating your soil, and in turn, you should see at least some earthworm activity.

Also, don't forget the weeds. Almost all of the plants humans label as weeds, have very deep sub-soil penetrating roots (most are also more nutritious than the stuff we've bred for us or other animals to eat). If you want to rebuild your soil, let the weeds grow and then either use them for compost which you'll later top-dress your soil with, or just chop them and leave them be for the cover crop effect, or work them in slightly, up to you.