im puzzled as to what you all are doing. doesn't the remark that b = sqrt(2)/a prove that the field Q(a) also contains b? hence is normal?
or am i way behind? certainly my earlier comments were incorrect, since i confused a biquadratic extension, namely one where both square roots adjoined were of elements in Q, with this situation where one adjoins the square root of 2, i.e. of an element of Q, and then a square root of an element of the larger field Q(sqrt(2)).
Indeed this is a standard way to get a non normal extension, generated by the real 4th roots of 2, for instance.
But if you want the galois group elements here, it seems they are:
1) identity
2) sqrt(2) fixed, hence a goes to the other root of its minimal polynomial over Q(sqrt(2)), namely -a. thus b goes to -b.
3) sqrt(2) goes to the other root of its minimal polynomial over Q, namely -sqrt(2). a goes to some root of its transformed minimal polynomial under the previous map on sqrt(2), say a goes to b. then since a field map takes quotients to quotients, b = sqrt(2)/a goes to -sqrt(2)/b = -a.
4) This one also takes sqrt(2) goes to the other root of its minimal polynomial over Q, namely -sqrt(2). Then a must go to the other root of its transformed minimal polynomial under the previous map, namely a goes to -b. then b = sqrt(2)/a goes to -sqrt(2)/-b = a.
the orders of these elements seem to be 1,2,4,4. In particular the group is cyclic of order 4.
the key observation being that of gonzo, that b = sqrt(2)/a.
I have not been able to follow all your discussion but to compute galois groups of an extension like Q(c,a) by hand, one simply sends c to some other root of its minimal polynomial over Q, then follows what happens to the minimal polynomial of a over Q(c) and sends b to some root of that transformed polynomial.
am i in the ball park? i admit i always was weak at algebra. which is why i wrote the book on my webpage. my book is full of thorough details, and shares everything i myself understand, but to understand things well i recommend reading a book by someone who gets it, like artin, or jacobson.
