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

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14 1. General extension fieldsTheorem 8. Let K/k be a transcendental extension generated by S andA a set of algebraically independent elements contained in S . Thenthere is a transcendence base B of K withA⊂ B⊂SProof. Since S is a set of generators of K, K/k(S ) is algebraic. Let Mbe the family of subsets A α of K with1) A⊂A α ⊂ S2) A α algebraically independent over k.□16The set M is not empty since A is in M. Partially order M by inclusion.Let{A α } be a totally ordered subfamily. Put B 0 = ⋃ αA α . ThenB o ⊂ S . Any finite subset of B o will be in some A α for largeαand soB o satisfies 2) also. Thus using Zorn’s lemma there exists a maximalelement B in M. Every element x of S depends algebraically on B forotherwise BU x will be in M and will be larger than B. Thus k(S )/k(B)is algebraic. Since K/k(S ) is algebraic, it follows that B satisfies theconditions of the theorem.The importance of the theorem is two fold; firstly that every set ofelements algebraically independent can be completed into a transcendencebase of K and further more every set of generators contains abase.We make the following simple observation. Let K/k be an extension,Z 1 ,...Z m , m elements of K which have the property that K/k(Z 1 ,...Z m )is algebraic, i.e., that Z 1 ,...,Z m is a set of generators. If Z∈Kthen Zdepends algebraically on Z 1 ,...,Z m i.e., k(Z, Z 1 ,...,Z m )/k(Z 1 ,...,Z m )is algebraic. We may also remark that if in the algebraic relation connectingZ, Z 1 ,...Z m , Z 1 occurs then we can say thatk(Z, Z 1 ,...Z m )/k(Z, Z 2 ,...,Z m )is algebraic which means that Z, Z 2 ,...Z m is again a set of generators.We now prove the
5. Transcendental extensions 15Theorem 9. If K/k has a transcendence base consisting of a finite numbern of elements, every transcendence base has n elements.Proof. Let Z 1 ,...,Z n and Z1 ′,...,Z′ m be two transcendence bases consistingof n and m elements respectively. If nm let n
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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: 5. Transcendental extensions 13Let
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 and 42: 6. Simple extensions 356) If k is n
Page 43 and 44: 6. Simple extensions 37Theorem 5. I
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
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2. Derivations 65which, by theorem
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3. Rational function fields 67Then
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3. Rational function fields 69Let z
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3. Rational function fields 71so th
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3. Rational function fields 73symbo
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Chapter 4Norm and Trace1 Norm and t
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1. Norm and trace 77We writef K/k (
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1. Norm and trace 79But{K : k}={K :
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2. Discriminant 81q n elements. The
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2. Discriminant 8395Theorem 4. Ifω
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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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