120 3. Factors and Zeros8.12. Use Exercise 2.12 to factor 0 = -w7 - x7 - y7 - z7.
8.13. Use Exercise 2.12. Note that 2(~~+y~+2~+xy+yt+zx) = z2+y2+z2.
Alternatively, set z = -(x + y) and simplify the three terms.
8.14. (a) Use the third equation to eliminate c from the other two.
8.16. If u is the desired quantity, then x3 - y3 + mz(x - y) = u(x2 - y2).
Note that x # y. Find a similar equation for y and z.
8.17. Any polynomial for which r is a zero must be divisible by P(x). Use
this fact to obtain an equation relating a and b.
8.19. (a) Th e si g ni fi cance of the condition that m is odd is that u -f- u #^0
for any element u in F. Consequently the equation 2x = k is always
solvable for 2. Choose h so that p(x + h) and q(x) have the same
linear coefficient, and then choose k.
(b) The reducible quadratics are easy to count-one for each pair of
zeros in F. Use (a) to show that the number of irreducibles is the
same for each linear coefficient of the quadratic.
8.21. Write f and g as a product of irreducibles and examine the exponents.
8.22. The first few cases are (f(x),g(x)) = (x, l), (2x2 - 1,2x), (4x3-3x,
4x2 - 1). Make a conjecture about f(x). Consult Exercise 1.3.8. Recall
that cos2 n0 - 1 = - sin2 nf?.
8.23. Write {m}! as a product of cyclotomic polynomials. What is the
exponent of &d(x)?
8.24. 2x/(1-x2) suggests a substitution x = tan u, etc. Look at tan(u+.. .).
Interpret the condition and the conclusion. Alternatively, put the left
side over a common denominator and express the numerator as a
polynomial in the elementary symmetric functions.
8.25. Multiply each di by two suitable expressions equal to 1 and add.
8.26. (a) If u is an imaginary zero of the quadratic, so is c = 21-l. What is
the sum and product of the zeros?
8.27. A second equation relating u and v is obtained by taking complex
conjugates. Note that iI= u-l, IY = 21-l.
8.28. Use de Moivre’s Theorem and the fact that sin 1712 = 1 to derive the
required equation.