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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