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Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite I am trying to show $mathbb Q(2+sqrt2)$ is isomorphic to $mathbb Q(sqrt2)$. Specifically, I have difficulty showing the first is contained in the latter. I must be missing something easy but I cannot seem to find a similar question that has been asked before. linear-algebra abstract-algebra share | cite | improve this question edited Sep 1 at 10:15 mathcounterexamples.net 25.6k 2 17 54 asked Sep 1 at 10:02 Homaniac 494 1 10 add a comment  |  up vote 2 down vote favorite I am trying to show $mathbb Q(2+sqrt2)$ is isomorphic to $mathbb Q(sqrt2)$. Specifically, I have difficulty showing the first is contained in the latter. I must be missing something easy but I cannot seem to find a similar question that has been asked before. linear-algebra abstract-algebra share | cite | improve this question edited Sep 1 at 10:15 mathcounterexampl

Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite Frank Drake proves in his 'large cardinals' book, pg 140, that Theorem: There are $kappa^+$ ordinals $alpha$ between $kappa$ and $kappa^+$ such that $alpha+1$ is not the 'constructive order' (i.e. constructive rank) of any subset of $kappa$. His proof makes use of the fact that there are $kappa^+$ models of ZFC in the form $L_alpha$ for some $kappalealphalekappa^+$. The proof looks fine to me, but I'm unsure about something. If no new subsets of $kappa$ are added from $L_alpha$ to $L_alpha+1$, then how exactly does the constructive hierarchy continue to grow? I mean, we cannot have $L_alpha = L_alpha+1 = L_alpha+2=...$ as then we would not get all of $L$. So when we move up the hierarchy, we must always be getting new sets from somewhere, but where exactly are they coming from? Furthermore, can we put an upper bound on the length of the gaps? (Drake shows using model theoretic arguments that