Visual Group Theory: Lecture 7.5: Euclidean domains and algebraic integers
8.9 هزار بار بازدید -
8 سال پیش
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Visual Group Theory: Lecture 7.5:
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 still works. It is not difficult to show that these rings must be PIDs. One common example are the Gaussian integers, which are the complex numbers with integer coefficients. A generalization of this are the algebraic integers, which is the subring algebraic numbers that are roots of monic polynomials in Z[x]. We are particularly interested in those that are contained in quadratic extension fields of the rationals, Q. In this lecture, we analyze these rings, and try to understand which are Euclidean and which are PIDs. Both of these complete classifications are still open, but we review what is known. We conclude with several pretty pictures of algebraic integers in the complex plane, and a updated diagram showing the relationships between the types of rings we've seen so far: fields, Euclidean domains, PIDs, UFD, integral domains, and commutative rings, and how they fit together.
Course webpage (with lecture notes, HW, etc.): http://www.math.clemson.edu/~macaule/...
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 still works. It is not difficult to show that these rings must be PIDs. One common example are the Gaussian integers, which are the complex numbers with integer coefficients. A generalization of this are the algebraic integers, which is the subring algebraic numbers that are roots of monic polynomials in Z[x]. We are particularly interested in those that are contained in quadratic extension fields of the rationals, Q. In this lecture, we analyze these rings, and try to understand which are Euclidean and which are PIDs. Both of these complete classifications are still open, but we review what is known. We conclude with several pretty pictures of algebraic integers in the complex plane, and a updated diagram showing the relationships between the types of rings we've seen so far: fields, Euclidean domains, PIDs, UFD, integral domains, and commutative rings, and how they fit together.
Course webpage (with lecture notes, HW, etc.): http://www.math.clemson.edu/~macaule/...
8 سال پیش
در تاریخ 1395/02/17 منتشر شده
است.
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