Sudhir R. Ghorpade2 - Lectures on Field Theory and...

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Unformatted text preview: Lectures on Field Theory and Ramification Theory Sudhir R. Ghorpade Department of Mathematics Indian Institute of Technology, Bombay Powai, Mumbai 400 076, India E-Mail: srg@math.iitb.ernet.in Instructional School on Algebraic Number Theory (Sponsored by the National Board for Higher Mathematics) Department of Mathematics, University of Bombay December 27, 1994 – January 14, 1995 Contents 1 Field Extensions 2 1.1 Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Basic Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Cyclic Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Abelian Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Ramification Theory 20 2.1 Extensions of Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Kummer’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Dedekind’s Discriminant Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 Ramification in Galois Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5 Decomposition and Inertia Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.6 Quadratic and Cyclotomic Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Bibliography 35 1 Chapter 1 Field Extensions 1.1 Basic Facts Let us begin with a quick review of the basic facts regarding field extensions and Galois groups. For more details, consult the notes [4] or any of the standard texts such as Lang [7] or Jacobson [6]. Suppose L/K is a field extension (which means that L is a field and K is a subfield of L ). We call L/K to be finite if as a vector space over K , L is of finite dimension; the degree of L/K , denoted by [ L : K ], is defined to be the vector space dimension of L over K . Given α 1 ,...,α n ∈ L , we denote by K ( α 1 ,... ,α n ) (resp: K [ α 1 ,...,α n ]) the smallest subfield (resp: subring) of L containing K and the elements α 1 ,... ,α n . If there exist finitely many elements α 1 ,... ,α n ∈ L such that L = K ( α 1 ,... ,α n ), then L/K is said to be finitely generated . An element α ∈ L such that L = K ( α ) is called a primitive element , and if such an element exists, then L/K is said to be a simple extension. If L ′ /K is another extension, then a homomorphism σ : L → L ′ such that σ ( c ) = c for all c ∈ K is called a K –homomorphism of L → L ′ . Note that a K –homomorphism is always injective and if [ L : K ] = [ L ′ : K ], then it is surjective. Thus if L = L ′ , then such maps are called K –automorphisms of L . The set of all K –automorphisms of L is clearly a group where the group operation defined by composition of maps. This is called thegroup operation defined by composition of maps....
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Sudhir R. Ghorpade2 - Lectures on Field Theory and...

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