An algebraic extension of fields $L|K$ is defined to be a *Galois extension* iff the set of elements of $L$ invariant under the action of $Aut\_K L$ is $K$. 

Apparently in the sequence of field extensions $L|M|K$ if $L|K$ is Galois then $L|M$ is. This is asserted to be an obvious deduction in section 3.2.1 of Robalo Delgados thesis on [Galois Categories](https://dspace.ist.utl.pt/bitstream/2295/575330/1/dissertacao.pdf) referenced in [nLab-Grothendiecks Galois Theory](http://ncatlab.org/nlab/show/Grothendieck's+Galois+theory), I don't see the obviousness...can someone clarify.

What interests me in this characterisation is that it shows one direction of the Galois correspondance, (the other being shown by Artins Lemma) without going through characterising Galois extensions as algebraic, seperable & normal extensions. In fact Delgado shows that this characterisation follows from his definition.

Edit: I've also asked this question on [math.overflow](https://mathoverflow.net/questions/120410/why-are-subextensions-of-galois-extensions-also-galois) as I thought most people misunderstood the question. I'd welcome attempts to formulate it in clearer terms.