Let L = C ( X , Y , Z ) be the rational function field over the complex field and σ be automorphism of L over C, σ ( X ) = Y , σ ( Y ) = Z , σ ( Z ) = XMoreover let M be the intermediate field of the extension L / C fixed by the group &lt; σ &gt;. I think the degree of filed extension L / M is 3 since the order of group &lt; σ &gt; is 3. But is it true? I know this is true if the extension L / C is finite degree Galois extension. But now the extension L / C is infinite degree extension. So I don't know it is true.Please give me some advice.

Question

Let   L  =      C    (  X  ,  Y  ,  Z  ) be the rational function field over the complex field and σ be automorphism of   L over   C,  σ  (  X  )  =  Y  ,  σ  (  Y  )  =  Z  ,  σ  (  Z  )  =  XMoreover let   M be the intermediate field of the extension   L      /        C   fixed by the group   &amp;lt;  σ  &amp;gt;. I think the degree of filed extension   L      /    M is 3 since the order of group   &amp;lt;  σ  &amp;gt; is 3. But is it true? I know this is true if the extension   L      /    C is finite degree Galois extension. But now the extension   L      /    C is infinite degree extension. So I don&#039;t know it is true.Please give me some advice.

Meadow Knox · Accepted Answer

Yes, it is a theorem that if   L is a field, and   G a finite group of automorphisms of   L, then   M  =      L    G  , the elements of   L fixed by all elements of   G, has the property that   L      /    M is a finite Galois extension with Galois group   G.See any text on Galois theory for the proof.

Let L = <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C </mrow>

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