Imagine computing by taking square and cube roots of real numbers, starting from some rational numbers. At every step of the process, we can consider the field generated by all of the numbers we have written down so far. So we get a sequence of fields with . Let . Let be the composite of and . One of the confusing things about this picture is that the are the fields which seem natural to concern ourselves with, as they involve the real numbers which we are actually computing with, but we need the to get the Galois theory to work out nicely.
Let be the Galois group of the normal closure of . Let be the subgroup of fixing and let be the subgroup fixing . Now, is obtained from by adjoining a cube or square root. We’ll deal with the cube root case first. In that case, is a -extension of , and an extension of . In other words, . The image of in is . If we have a square root extension, then , with the image of generating one factor.
We want to use the assumption that lies in . So . The extension is Galois, with Galois group generated by . Probably the easiest way to see this Galois group is to notice that lies in , which is Galois over with Galois group the unit group of . In any case, the assumption that lies in means that there is a surjection , whose kernel contains .
Consider the restriction of to . Since is trivial on , we get a surjection . But or and there are no nontrivial maps from or to . So dies on and, in particular, on . Now, we can repeat the argument, looking at . Continuing inductively, we deduce that is trivial on all of . But is supposed to be a surjection, a contradiction. QED
One of these days I’ll get a chance to teach Galois theory. There are so many pretty little examples like this, and so few of them find their way into textbooks.