Commutative algebra

Integrally closed domain

In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in its field of fractions is A itself. Spelled out, this means that if x is an element of the field of fractions of A which is a root of a monic polynomial with coefficients in A, then x is itself an element of A. Many well-studied domains are integrally closed: fields, the ring of integers Z, unique factorization domains and regular local rings are all integrally closed. Note that integrally closed domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields (Wikipedia).

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👉 Learn about integration. The integral, also called antiderivative, of a function, is the reverse process of differentiation. Integral of a function can be evaluated as an indefinite integral or as a definite integral. A definite integral is an integral in which the upper and the lower li

From playlist The Integral

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From playlist The Integral

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From playlist The Integral

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Integral Domains are essentially rings without any zero divisors. These are useful structures because zero divisors can cause all sorts of problems. They complicate the process of solving equations, prevent you from cancelling common factors in an equation, etc. In this lesson we intro

From playlist Abstract Algebra

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From playlist The Integral

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From playlist Evaluate Integrals

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From playlist The Integral

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From playlist The Integral

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From playlist Commutative algebra

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From playlist Mathematics

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This video explores contour integration of functions in the complex plane. @eigensteve on Twitter eigensteve.com databookuw.com

From playlist Engineering Math: Crash Course in Complex Analysis

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From playlist Complex Analysis

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Complex Analysis - Part 23 - Cauchy's Theorem (for discs)

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Related pages

Field of fractions | Prime ideal | Integral domain | Subclass (set theory) | Valuation ring | Regular local ring | Associated prime | Maximal ideal | GCD domain | Alexander Grothendieck | Minimal polynomial (field theory) | Commutative algebra | Unibranch local ring | Integral element | Principal ideal domain | Field extension | Dedekind domain | Algebraic element | Monic polynomial | Field (mathematics) | Bézout domain | Localization (commutative algebra) | Noetherian ring | Reduced ring | Square-free polynomial | Going up and going down | Discrete valuation ring | Irreducible polynomial | Singular point of a curve | Unique factorization domain | Symmetric algebra | Fractional ideal | Normal scheme