By Alvin K Bettinger

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**Example text**

Dihedral) group embeds into PGL2(k) (since k is algebraically closed of characteristic 0). The other finite subgroups ofPGL2(k) are only A4, S4 and A5; and any two finite subgroups of PGL2(k) that are isomorphic as abstract groups are conjugate in PGL2(k). This was proved by Klein in 1876 (see [Klein] and for a modem presentation, see Lamotke [Lam]). (c) Deduce from (a) and (b) that if k(y)/ k(x) is a Galois extension then its group is either cyclic, dihedral or isomorphic to A4, S4 and A 5. By Liiroth's theorem, all those cases actually occur.

Now assume that feb, X2, ... , x s ) is reducible, say feb, X2, ... , x s ) = h(X2, ... , x s )h'(X2, ... , xs), where hand h' are both not constant. The Kronecker specializations Sdh(y) and Sdh'(y) are defined similarly as above. We have Sdf(b, y) = Sdh(y)Sdh'(y), hence Sdh(y) and Sdh'(y) are each a product of certain gi (b, y) (up to factors from k). Let H (x, y) and H' (x, y) be the product of the corresponding gj(x, y). Then Sdf(x, y) = g(x)H(x, y)H'(x, y). Because of the uniqueness of the d-adic expansion of an integer, there are unique polynomials h(xJ, ...

Exercise Galois extensions k(y) / k(x). The simplest case of an extension L / k(x) is that where L = k(y) is itself a rational function field. If this extension is Galois then its group G embeds into the group Aut(k (y) / k). Deduce that the group Aut(k(y)/ k) is isomorphic to PGL2(k) (the two-dimensional projective linear group over k). , dihedral) group embeds into PGL2(k) (since k is algebraically closed of characteristic 0). The other finite subgroups ofPGL2(k) are only A4, S4 and A5; and any two finite subgroups of PGL2(k) that are isomorphic as abstract groups are conjugate in PGL2(k).