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Math Help - galois theory

  1. #1
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    galois theory

    Can someone pls pls pls help me with this questions asap! Thanks so much

    Edgar

    Let L = C(s,t) where s and t are indeterminates (or more formally L is the field of fractions of C[s,t]). Let Φ: L → L be given by Φ(s) = t, Φ(t) = s, and let G be the group generated by Φ. Find the fixed field of G, justifying your answer.
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  2. #2
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    Quote Originally Posted by edgar davids View Post
    Let L = C(s,t) where s and t are indeterminates (or more formally L is the field of fractions of C[s,t]). Let Φ: L → L be given by Φ(s) = t, Φ(t) = s, and let G be the group generated by Φ. Find the fixed field of G, justifying your answer.
    I never studied anything about these types of problems before so this is just a guess.

    Say that,
    \alpha = a_0^0 + a_1s+...+a_n^0 s^n
     a_0^1t+a_1^1 ts+...+a_n^1 ts^n
    ....
    a_0^n t^n + a_1^n t^n s+..+a_n^n t^ns^n .

    It must have this form because, it cannot be that degree of t exceedes degree of s for that would make the polynomial of degree s exceedes degree of t which means it cannot be the same after applying the automorphism \theta to it.

    Now this value stays unchanged under \theta. The values across the diagnols can be anything while a_k^j = a_j^k.

    For example,
    2+s+6s^2
    t+5st+3s^2t
    6t^2+3t^2s+9t^2s^2

    We can pair it as,
    2+(s+t)+6(s^2+t^2)+ 5st+3st(s+t)+9(st)^2.

    This seems to suggest that,
    L^G = C(st,s+t,s^2+t^2,s^3+t^3,...)

    But it turns out that C(st,s+t.s^2+t^2,...) = C(st,s+t).

    Because s^n+t^n can be obtained from st,s+t for n\geq 2. For example, s^2+t^2 = (s+t)^2 - 2st. And s^3+t^3 = (s+t)^3 - 3st(s+t). And so on.

    Thus. the fixed field under the group G is L^G = C(st,s+t).
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