I am looking to put a section together involving the Hasse Norm Theorem in a piece of work I am writing. As well as the Hasse Norm Theorem itself, Wikipedia also mentions a theorem by Hilbert in 1897 for the special case n = 2 and the case n is prime by Furtwangler 1902. I was wondering if anybody could point me in the direction of any books or online resources concerning these theorems and maybe even their proofs. I have also seen two well known counter examples namely,
$\mathbb{Q}(\sqrt{-3},\sqrt{13})/\mathbb{Q}$ and $\mathbb{Q}(\sqrt{13},\sqrt{17})/\mathbb{Q}$,
by Hasse and Serre, Tate respectively. I would also be grateful if anybody could highlight where I may find literature on these counterexamples.
Hilbert's work is contained in his collected papers, vol I, right after the Zahlbericht. Furtwängler's proofs are available online (see https://www.emis.de/MATH/JFM/ for finding the references). Their approach is highly computational; it took me a very long time to figure out the ideas behind them. But all of these articles are available online, if you want to have a look at them.
The counterexamples you mention were studied in detail by Arnold Scholz; search for Scholz and "number knots" to find modern literature; the standard reference is Jehne's article "On knots in algebraic number theory" (see https://eudml.org/doc/152174). I also recommend my notes https://www.mathi.uni-heidelberg.de/~flemmermeyer/publ/pcft.pdf and some sections in the book "Der Briefwechsel Hasse - Scholz - Taussky" (https://univerlag.uni-goettingen.de/handle/3/isbn-978-3-86395-253-2).