How to show a theory which is categorical of an uncountable cardinality is totally transcendental?

I am studying Morley's theorem, which says that a complete theory which is countable and categorical in an uncountable cardinality $$\kappa$$ is categorical in every uncountable cardinality. In order to prove this we claim that if a theory is categorical in an uncountable cardinality, then it is totally transcendental. But I am stuck here, and I can not prove it.

I tried to show instead it is $$\omega$$-stable by contadiction: Let $$M$$ be a model of $$T$$ with $$|M|= \aleph_0$$ but $$|S_1(M)| \geq \aleph_1$$. By the Löwenheim-Skolem theorem, there exists an elementary extension $$M\preceq N$$ with $$|N|=\kappa$$, but I can not find a way to contradict with $$\kappa$$-categoricity...

• I cleaned up your post, fixing lots of spelling, grammar, and formatting errors. Please try to pay more attention to these things in the future. – Alex Kruckman Mar 8 '19 at 14:41
• Also, I changed "there exists an initial expansion of $m$ like $n$" to "there exists and elementary extension $M\preceq N$"... I have no idea what an initial expansion is, but I guessed from context that this is what you meant. I hope I guessed correctly! – Alex Kruckman Mar 8 '19 at 14:47
• Yes,thanks a lot...I really have to try to improve my english..... – user297564 Mar 8 '19 at 15:13

You have the right idea to prove that the theory is totally transcendental by proving that it is $$\omega$$-stable. But this is a non-trivial component of Morley's theorem, and one that you're not going to be able to prove just by playing around with Löwenheim-Skolem.
In any case, the idea is this: You show that for every every countable theory $$T$$ with infinite models and every infinite cardinal $$\kappa$$, there is a model $$M\models T$$ which only realizes countably many types over any countable set of parameters. How? You build an Ehrenfeucht-Mostowski model generated by a sequence of indiscernibles ordered like the ordinal $$\kappa$$ (this is the tricky part of the proof, which would take far too long to explain in this answer). But on the other hand, if there is a countable model $$N$$ of $$T$$ such that $$|S_1(N)|\geq \aleph_1$$, then you can realize a set of $$\aleph_1$$-many types over $$N$$ in an elementary extension $$N\preceq N'$$ of size $$\kappa$$. These two models $$M$$ and $$N'$$ aren't isomorphic, so $$T$$ isn't $$\kappa$$-categorical.