Hans Schoutens (CityTech and Graduate Center) - Ultraproducts in algebra - October 31, 2011

<p>
 Ultraproducts are some sort of infinite quantum computers:
they make a new algebra structure from a given infinite collection of
structures, by defining the new operations by some random, whence
generic,  choices. Thus the ultraproduct of fields is again a field, the
ultraproduct of local rings is again a local ring, etc. Rather than
giving a technical treatment, I will try to convince you with examples.
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This allows us to generate the field of complex numbers as an
ultraproduct of a limit of finite fields, the so-called  weak Lefschetz
Principle. The advantage of working in finite fields (apart from the
fact that computers can only store finitely many bits!) is that  they
admit a simplified binomial theorem. Lifting this to the complexes
yields the "students' binomial theorem" $(x+y)^a=x^a+y^a$ (but the
exponent $a$ is of course very special, a so-called ultraprime element).