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Consider the poset of number fields, partial order being defined by inclusion of fields. What is the group of order-preserving automorphisms of this poset? What if we take only Galois extensions of $\mathbb{Q}$?

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    $\begingroup$ You mean, you fix an algebraic closure and consider the poset of number subfields inside it. $\endgroup$
    – YCor
    Commented Apr 12, 2019 at 7:35
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    $\begingroup$ If you consider a finite field $\mathbb F_q$ in place of $\mathbb Q$, then the lattice is isomorphic to the lattice of integers under divisibility. The automorphisms of that lattice are generated precisely by permutations of primes, so the automorphism group is isomorphic to the symmetric group on infinitely many elements. A similar approach probably won't work for $\mathbb Q$ though... $\endgroup$
    – Wojowu
    Commented Apr 12, 2019 at 7:43

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