Lectures on the Algebraic Theory of Fields - Tata Institute of ...

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36 2. Algebraic extension fields42Putγ=β 1 +λα 1 =β+λαObviouslyλcan be chosen so thatγ0Now k(γ)⊂k(α,β). We shall will now show thatα∈k(γ). Fromdefinition ofγ,βalso will be in k(γ) and that will meank(γ)⊂k(α,β)⊂k(γ).In order to do this consider the polynomialφ(γ−λ.x)in k(γ)[x]. Alsoit vanishes for x=α. Furthermore by our choice ofλ,φ(γ−λα i )0for i>1. In the algebraic closureΩ, therefore, f (x) andφ(γ−λx) havejust x−α as a factor. But f (x) andφ(γ−λx) are both polynomials ink(γ)[x]. So x−αis the greatest common divisor ofφ(γ−λx) and f (x)in k(γ)[x]. Thusα∈k(γ). Our lemma is demonstrated.We have thereforeCorollary . Ifα 1 ,...,α n are separably algebraic and β ∈ Ω thenk(β,α,...α n ) is a simple extension.We deduce immediatelyCorollary. A finite separable extension is simple.LetΓbe the field of rational numbers andΓ(ω,ρ) the splitting fieldof the polynomial x 3 − 2 inΓ[x]. ThenΓ(ω,ρ) is simple. A primitiveelementγis given byω+ρ=γ. ThenΓ(γ) is of degree 6 overΓ. It iseasy to see thatγhas overΓthe minimum polynomial(x 3 − 3x−3) 2 + 3x(x+1)(x 3 − 3x−3)+9x 2 (x+1) 2Let now K/k be a finite extension and L the maximal separablesubfield of K/k. Then (K : L)= p f the degree of inseparability and(L : k)=d the reduced degree. If we consider the exponents of elementsof K, these have a maximum e ande≤ f.43We had given an example of e< f . We shall now prove the
6. Simple extensions 37Theorem 5. If e is the exponent and p f the degree of inseparability offinite extension K of k, then e= f⇐⇒ K/k is simple.Proof. Let K= k(ω). Let K o be the maximal separable subfield of K/k.Now (K : K o )= p f . But p f is degree ofω. Thus e= f □Let now K/k be a finite extension and e= f . There exists then aωin K such thatω pe is separable and in K o but for no, t
Page 2 and 3: Lectures on theAlg
Page 5: Contents1 General extension fields
Page 15 and 16: 4. Algebraic Closure 9thatτ·σ is
Page 17 and 18: 4. Algebraic Closure 11¯σx=σ α
Page 19 and 20: 5. Transcendental extensions 13Let
Page 21 and 22: 5. Transcendental extensions 15Theo
Page 23 and 24: Chapter 2Algebraic extension fields
Page 25 and 26: 2. Normal extensions 19for all auto
Page 27 and 28: 3. Isomorphisms of fields 213 Isomo
Page 29 and 30: 3. Isomorphisms of fields 23mapping
Page 31 and 32: 4. Separability 25For, the minimum
Page 33 and 34: 4. Separability 27f (x). Thenφ(y)
Page 35 and 36: 4. Separability 29From the definiti
Page 37 and 38: 5. Perfect fields 31has characteris
Page 39 and 40: 5. Perfect fields 335) Any algebrai
Page 41: 6. Simple extensions 356) If k is n
Page 45 and 46: 7. Galois extensions 39Proof. Let k
Page 47 and 48: 7. Galois extensions 41σ −1 τσ
Page 49 and 50: 7. Galois extensions 43From linear
Page 51 and 52: 7. Galois extensions 45Letσ ∈ G(
Page 53 and 54: 8. Finite fields 471 being the unit
Page 55 and 56: Chapter 3Algebraic function fields1
Page 57 and 58: 1. F.K. Schmidt’s theorem 51both
Page 59 and 60: 1. F.K. Schmidt’s theorem 53Thus
Page 61 and 62: 2. Derivations 55so De=o. If Da=o,
Page 63 and 64: 2. Derivations 57then write 1 i = a
Page 65 and 66: 2. Derivations 59aλ 1 ,...,λ m
Page 67 and 68: 2. Derivations 61we getf D (x)+m∑
Page 69 and 70: 2. Derivations 63If we take K to be
Page 71 and 72: 2. Derivations 65which, by theorem
Page 73 and 74: 3. Rational function fields 67Then
Page 75 and 76: 3. Rational function fields 69Let z
Page 77 and 78: 3. Rational function fields 71so th
Page 79 and 80: 3. Rational function fields 73symbo
Page 81 and 82: Chapter 4Norm and Trace1 Norm and t
Page 83 and 84: 1. Norm and trace 77We writef K/k (
Page 85 and 86: 1. Norm and trace 79But{K : k}={K :
Page 87 and 88: 2. Discriminant 81q n elements. The
Page 89 and 90: 2. Discriminant 8395Theorem 4. Ifω
Page 91 and 92: 2. Discriminant 85D K/k (ω)=(−1)
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Chapter 5Composite extensions1 Kron
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1. Kronecker product of Vector spac
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1. Kronecker product of Vector spac
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2. Composite fields 93The elements
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2. Composite fields 95109Letϕbe th
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3. Applications 97Since L i is a fi
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3. Applications 993) If (K : k)=m a
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3. Applications 101over K∩ L and
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Chapter 6Special algebraic extensio
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2. Cyclotomic extensions 105Any x i
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2. Cyclotomic extensions 107automor
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2. Cyclotomic extensions 109If m =
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2. Cyclotomic extensions 111Alsox 4
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3. Cohomology 1133 CohomologyLet G
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3. Cohomology 115Therefore, the ele
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3. Cohomology 117Theorem 6. H o (G,
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4. Cyclic extensions 1194 Cyclic ex
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4. Cyclic extensions 121139We study
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4. Cyclic extensions 123elements of
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4. Cyclic extensions 125in L[x]. Le
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5. Artin-Schreier theorem 127where
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6. Kummer extensions 129α n is the
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6. Kummer extensions 131Observe, no
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7. Abelian extensions of exponent p
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8. Solvable extensions 135over k.k(
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8. Solvable extensions 137Lemma 2.
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8. Solvable extensions 139the alter
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8. Solvable extensions 141Thenω 3
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8. Solvable extensions 143we now ha
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8. Solvable extensions 145y 1 , y 2
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8. Solvable extensions 147Suppose,
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150 7. Formally real fields171non-n
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152 7. Formally real fields2 Extens
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154 7. Formally real fields176β i
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156 7. Formally real fields3 Real c
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158 7. Formally real fieldsConsider
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160 7. Formally real fieldsAll the
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162 7. Formally real fieldsis true
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164 7. Formally real fields3) In or
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166 7. Formally real fields4 Comple
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168 7. Formally real fields(2) ¯k
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170 7. Formally real fieldslim a n
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172 7. Formally real fields198In or
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Chapter 8Valuated fields1 Valuation
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2. Classification of valuations 177
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2. Classification of valuations 179
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3. Examples 181Replace n by n r , w
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3. Examples 183208We shall denote t
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4. Complete fields 185210The Cauchy
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4. Complete fields 187Now a n = a n
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4. Complete fields 189215Proof. The
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4. Complete fields 1916) Every elem
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4. Complete fields 193where f (x)
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5. Extension of the valuation of a
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6. Fields complete under archimedia
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6. Fields complete under archimedia
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7. Extension of valuation of an inc
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7. Extension of valuation of an inc
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210 8. AppendixLemma 3. Let G be a
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212 8. Appendix241242On the other h
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214 8. AppendixIf we denote the uni
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216 8. Appendix246247where b i is a
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218 8. Appendixfor allσ∈G. Then
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Bibliography[1] A. A Albert Modern
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