Tag Archives: finite extensions

Commutative Algebra 42

Posted on April 30, 2020 by limsup

Noether Normalization Theorem Throughout this article, k is a field, not necessarily algebraically closed. Definition. Let A be a finitely generated k-algebra which is an integral domain. We say are algebraically independent over k if they are so as elements … Continue reading →

Posted in Advanced Algebra | Tagged , , , , , | 4 Comments

Commutative Algebra 40

Posted on April 28, 2020 by limsup

More on Integrality Lemma 1. Let be an integral extension. If is an ideal and , the resulting injection is an integral extension. Proof Any element of can be written as , . Then x satisfies a monic polynomial relation: . … Continue reading →

Posted in Advanced Algebra | Tagged , , , , , , | 4 Comments

Commutative Algebra 39

Posted on April 27, 2020 by limsup

Integrality Throughout this article, A is a subring of B; we will also call B a ring extension of A. Definition. An element is said to be integral over A if we can find (where ) such that in B. For example, is integral over since … Continue reading →

Posted in Advanced Algebra | Tagged , , , , , , | 4 Comments