OP827 wrote:Omhs law: V=I*R. Provided R is constant then if V drops 10 times then I (current) will also drop 10 times.
I agree that's basically correct, BUT the curveball is that R is not constant.
The conductivity of metals generally decreases with temperature.
If you put in only 12 volts, the element doesn't get as hot and the current flow at 12V is more than 1/10 of the current at 120V. Assuming a metal element, of course.
And, of course, impedance to an AC supply isn't the same as resistance to a DC supply, as Scott points out.
I vote get a 12V blanket or mattress pad, too.