Square Root as a Homomorphism

  title={Square Root as a Homomorphism},
  author={William C. Waterhouse},
  journal={The American Mathematical Monthly},
In any field K where−1 is not a square, there are homomorphisms from the squares to (chosen) square roots in K . Maximal fields F with such homomorphisms have Gal(F̄/F) either Z/2Z or the 2-adic integers. By and large, high school algebra students (thinking of positive real numbers) learn that √ cd = √ c · √ d . Most mathematicians know that there is no way of making this work for complex numbers. The most common account of this (see [3], for instance) seems to focus on how a square root… CONTINUE READING