461hw6 - why f ( a,b,c,d ) is a real number. (Hint: The...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 461 F Spring 2011 Homework 6 Drew Armstrong Reading. Section 6.4, 6.5 Book Problems. 6.4: 1, 4, 8. Additional Problems. A.1. Euler showed that every real polynomial of the form x 4 + αx 2 + βx + γ factors into two real quadratics. Use his result to prove that every real polynomial of the form ax 4 + bx 3 + cx 2 + dx + e factors into two real quadratics. (Hint: Use a change of variables to turn your polynomial into Euler’s polynomial.) A.2. Suppose that a given real quartic equation has roots a,b,c,d in some field E R . (Today we know that these roots must be complex, but in times past their nature was mysterious.) Now let f ( a,b,c,d ) be some function of a,b,c,d that is invariant under any permutation of the roots. Explain
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: why f ( a,b,c,d ) is a real number. (Hint: The Fundamental Theorem of Symmetric Functions.) A.3. Let a,b,c,d be the roots of some real quartic equation with no x 3 term (i.e. we have a + b + c + d = 0.) Let p = a + b , q = a + c , and r = a + d , so that-p = c + d ,-q = b + d , and-r = b + c . Prove that pqr is a real number, and hence-p 2 q 2 r 2 is a negative real number. (Hint: Show that pqr is invariant under permuting a ↔ b , or a ↔ c , or a ↔ d . Hence it’s invariant under any permutation of a,b,c,d .)...
View Full Document

This note was uploaded on 01/08/2012 for the course MATH 561 taught by Professor Armstrong during the Spring '11 term at University of Miami.

Ask a homework question - tutors are online