Commutative algebra | Ring theory
In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal generated by two given elements. Equivalently, any two elements of R have a least common multiple (LCM). A GCD domain generalizes a unique factorization domain (UFD) to a non-Noetherian setting in the following sense: an integral domain is a UFD if and only if it is a GCD domain satisfying the ascending chain condition on principal ideals (and in particular if it is Noetherian). GCD domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields (Wikipedia).
A review of the notes common to all formations of a G chord.
From playlist Music Lessons
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From playlist The Internet
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From playlist The Internet
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Visual Group Theory: Lecture 7.5: Euclidean domains and algebraic integers
Visual Group Theory: Lecture 7.5: Euclidean domains and algebraic integers. Around 300 BC, the Greek mathematician Euclid found an algorithm to compute the greatest common divisor (gcd) of two numbers. Loosely speaking, a Euclidean domain is a commutative ring for which this algorihm stil
From playlist Visual Group Theory
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From playlist The Internet
Visual Group Theory: Lecture 7.4: Divisibility and factorization
Visual Group Theory: Lecture 7.4: Divisibility and factorization The ring of integers have a number of properties that we take for granted: every number can be factored uniquely into primes, and all pairs of numbers have a unique gcd and lcm. In this lecture, we investigate when this happ
From playlist Visual Group Theory
Abstract Algebra | Eisenstein's criterion
We present a proof of Eisenstein's criterion along with some examples. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Personal Website: http://www.michael-penn.net Randolph College Math: http://www.randolphcollege.edu/mathematics/ Research Gate profile: htt
From playlist Abstract Algebra
From playlist Abstract Algebra 2
Abstract Algebra | If D is a UFD then D[x] is a UFD.
We prove an important result that states the ring of polynomials whose coefficients are from a unique factorization domain is itself a unique factorization domain. Along the way, we define the content of a polynomial, prove Gauss' lemma, and prove that if a polynomial factors over the fiel
From playlist Abstract Algebra
MAG - Lecture 2 - Ideals and polynomial division
metauni Algebraic Geometry (MAG) is a first course in algebraic geometry, in Roblox. In Lecture 2 we introduce ideals, the affine variety associated to an ideal, the ideal of an affine variety, and the polynomial division algorithm along with several examples. The webpage for MAG is https
From playlist MAG
A Short Course in Algebra and Number Theory - Elementary Number Theory
To supplement a course taught at The University of Queensland's School of Mathematics and Physics I present a very brief summary of algebra and number theory for those students who need to quickly refresh that material or fill in some gaps in their understanding. This is the fourth lectu
From playlist A Short Course in Algebra and Number Theory
Abstract Algebra | Writing a polynomial gcd as a combination -- example.
We give an example of Bezout's identity in polynomials. This involves the extended Euclidean algorithm for polynomials. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Personal Website: http://www.michael-penn.net Randolph College Math: http://www.randolphcol
From playlist Abstract Algebra
Mark Giesbrecht 4/23/15 Part 1
Title: I. Approximate Computation with Differential Polynomials: Approximate GCRDs II. Sparsity, Complexity and Practicality in Symbolic Computations Symbolic-Numeric Computing Seminar
From playlist Symbolic-Numeric Computing Seminar
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Cryptography and Network Security by Prof. D. Mukhopadhyay, Department of Computer Science and Engineering, IIT Kharagpur. For more details on NPTEL visit http://nptel.iitm.ac.in
From playlist Computer - Cryptography and Network Security