Visual Group Theory, Lecture 6.8: Impossibility proofs
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8 سال پیش
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Visual Group Theory, Lecture 6.8:
Visual Group Theory, Lecture 6.8: Impossibility proofs
The ancient Greeks sought basic ruler and compass constructions such as (1) squaring the circle, (2) doubling the cube, and (3) trisecting an angle. In the previous lecture, we learned how a length or angle 'z' is constructable iff the degree [Q(z):Q] is a power of 2. In this lecture, we will rephrase these three classic constuctions in terms of field extensions, and show that they are impossible. We will also learn about how the 19th century mathematicians Gauss and Wantzel solved an outstanding problem from the ancient Greeks about constructing regular n-gons. Their theorem says that a regular n-gon is constructible if and only if n is a product of 2^k and distinct Fermat primes. A Fermat prime is a prime of the form p=2^{2^n}+1, and so far, only 5 are known to exist.
Course webpage (with lecture notes, HW, etc.): www.math.clemson.edu/~macaule/math4120-online.html
8 سال پیش
در تاریخ 1395/02/08 منتشر شده
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