The Galois group of F(t)

Let F be a field and t an indeterminate. By F (t) we denote the field

F (t) = { fg

:f,g polynomials in t and g 6= 0}.

1. Let α = a(t)b(t) , with a and b coprime in F (t). Then

(a) Element α is transcendental over F.

(b) The polynomial a(X) − αb(X) ∈ F (α)[X] is irreducible over F (α).

(c) F (α) ⊂ F (t) is a finite extension of degree |F (t) : F (α)| =max(deg(a),deg(b)).

Let GL(2, F ) be the group of 2 × 2 invertible matrices with entries in F. To( a b

c d

)

∈ GL(2, F )

we associate the element at+bct+d

∈ F (t).

2. (a) For γ =( a b

c d

)

∈ GL(2, F ) the function σγ : α(t) → α(at+bct+d

) is an element

of Gal(F (t)/F ).

(b) The map γ → σγ−1 is a group homomorphism from GL(2, F ) to Gal(F (t)/F )that is surjective and has as kernel F ∗I, the nonzero multiples of the identity matrix.

(c) Gal(F (t)/F ) is isomorphic to the projective general linear group PGL(2, F ) =GL(2, F )/F ∗I.

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