Galois Theory: A Study of Cyclotomic Field ... - Scripps College

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24 Cyclotomic Field Extensions Q(ζ 19 ) 3 2 Q(ζ + ζ 7 + ζ 11 ) Q(ζ + ζ −1 ) 3 2 3 Q(ζ + ζ 4 + ζ 5 + ζ 6 + ζ 7 + ζ 9 + ζ 11 + ζ 16 + ζ 17 ) 2 Q(ζ + ζ 7 + ζ 8 + ζ 12 + ζ 18 ) 3 Q Figure 4.2: Field Lattice for Q(ζ 19 ) degree 2 and degree 3 extensions from Q to Q(ζ 19 ). Fortunately, this is not all that is known about the subfields. In the following section, we will get a better understanding of these two intermediate fields. 4.5 Real Subfields and Quadratic Extensions Proposition 17. The subfield L max of K = Q(ζ p ) whose degree over Q is 1 2 (p−1) is generated over Q by the element η = ζ +ζ p−1 = 2 cos 2π/p. Moreover, L max = K ∩ R, so L max is called the maximal real subfield of K. Proof. Consider the quadratic polynomial x 2 − ηx + 1, which has coefficients in Q(η) and η = ζ + ζ −1 . Note that ζ is a root. So we know that
Real Subfields and Quadratic Extensions 25 [K : Q(ζ)] ≤ 2. Since η is a real number and ζ is not, Q(η) ⊂ K. So, [K : Q(η)] = 2, and Q(η) = K ∩ R. Thus, [Q(η) : Q] = 1 2 (p − 1). □ Thus in our example of Q(ζ 19 ), we can simplify our representation of our real subfield of Q(ζ 19 ): ( ) However, if 2 cos 2π 19 ( ( )) 2π Q(ζ + ζ −1 ) = Q 2 cos . 19 ( ) is in L max , then certainly cos 2π 19 will also be in the field. So we can simplify our representation of the real subfield further by ( ( )) simply writing it as L max = Q cos 2π 19 . The maximal real subfield is just a degree 2 subfield of Q(ζ 19 ). In other words, [Q(ζ 19 ) : L max ] = 2. 4.5.1 Discriminant of a Quadratic Extension Given any f(x) ∈ Q[x] and a splitting field K, we write f(x) = (x − α 1 ) · · · (x − α n ). Since K is a splitting field for f(x), all the roots of f(x) are going to be in K. Note that any σ ∈ G(K/Q) permutes these roots. Definition. The discriminant of a polynomial f(x) = (x − α 1 )(x − α 2 ) · · · (x − α n ) is defined to be D = (α 1 − α 2 ) 2 (α 1 − α 3 ) 2 · · · (α n−1 − α n ) 2 = ∏ (α i − α j ) 2 = ± ∏ i − α j ). i
Page 1: Galois Theory: A Study of Cyclotomi
Page 5: Contents Abstract Acknowledgments i
Page 9: List of Tables 4.1 Field Generator
Page 13: Chapter 1 Introduction Galois theor
Page 16 and 17: 4 The Basics of Field Theory Defini
Page 19 and 20: Chapter 3 Galois Theory 3.1 Introdu
Page 21 and 22: Introduction 9 is, L = L ′ = K. T
Page 23 and 24: The Main Theorem 11 L ⊂ K, we kno
Page 25 and 26: Proof of the Main Theorem of Galois
Page 27: Intermediate Galois Extensions 15 i
Page 30 and 31: 18 Cyclotomic Field Extensions grou
Page 32 and 33: 20 Cyclotomic Field Extensions a =
Page 34 and 35: 22 Cyclotomic Field Extensions Then
Page 38 and 39: 26 Cyclotomic Field Extensions Q co
Page 40 and 41: 28 Cyclotomic Field Extensions that
Page 42 and 43: 30 Cyclotomic Field Extensions sion
Page 45: Bibliography Artin, M. (1991). Alge

Galois Theory: A Study of Cyclotomic Field ... - Scripps College

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