## Elements of algebraic geometry: lectures |

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### Contents

Valuation Rings Places and Valuations | 32 |

Absolutely Irreducible Varieties | 63 |

Projective Varieties | 103 |

1 other sections not shown

### Common terms and phrases

absolutely irreducible alge algebraic geometry algebraic points algebraically closed algebraically free algebraically independent ascending chain condition assume automorphism belong called closure coefficients common non-trivial solution component varieties concept consequently consists containing degree of transcendency denote dimension equation equivalent generic points exists extension F and k^p F are linearly field of definition finite number finitely generated subfield form of degree form of positive Hence Hilbert Basis Theorem homogeneous algebraic set homogeneous varieties homomorphism ideal defined ideal of k[X implies integrally closed intersection isomorphism kernel largest defining ideal lemma let F let us show linear linearly disjoint linearly independent maximal ideal Modern Algebra monomials Noetherian Noetherian ring non-units non-zero obtain polynomial positive degree prime ideal projective varieties Proof prove quotient field resultant satisfied specialization subset subvariety Suppose Theorem 3.1 transcendence base trivial valuation ring variables well-defined whence wish to show