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What does this mean in the context of Galois theory?

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    • Thread Starter

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    I tried to highlight the bit of terminology I don't understand and give some context to this.

    The bits I don't get are mainly the "action" (unless I'm getting it correct below as a guess of what I think it could mean) and "(element wise) stabilisers" and "stabilisers", if anyone could explain these I would be grateful as I don't see what it means.

    In this particular case we are talking about a group of automorphisms acting on a set of elements of a field. I believe the "action" of G on the field is just applying an element of G (an automorphism) to a element of the field. (Is this correct?) The group here is the Galois group of some arbitrary field extension L/K and the field it acts on is L.

    But I have no idea what the highlighted paragraph means in particular I have never come across the term elementwise stabiliser? Could it mean the set of all automorphisms of G that map each element of {u1,...,uk} into {u1,...,uk} i.e. some kind of permutation on these k elements? Or is it something else entirely?

    I will answer any questions if I haven't been clear enough in due course.
    • Thread Starter

    EDIT: I believe I understand now. Set-wise is the set of all automorphisms in Gal(L/K) that map {u1,...,uk} into the set {u1,...,uk} and element-wise is the set of all automorphisms in Gal(L/K) that map each ui in {u1,...,uk} to itself.
    • Thread Starter

    In particular I think the highlighted bit is the set

    \displaystyle \{\Phi \in \text{Gal}(L/K): \Phi (u_i)=u_i ~\forall i=1,2,...,k\}

    any confirmation on this would be nice though.

    (Original post by matthew69238)
    Any confirmation on this would be nice though.
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