1.8. Problems on Quadratics 39(b) Show that Z, is a commutative ring with identity.
(c) Characterize those values of m for which Z, is a field.
(d) Show that, if Z, is not a field, then it is not even an integral
domain.- Write down a complete list of polynomials of degrees 0, 1, 2, 3, 4 in
the ring Z,[t]. Indicate in your list which of the polynomials cannot
be obtained by multiplying two polynomials of lower degree. This will
include all polynomials of degrees 0 and 1. Will it also include any
polynomials of degrees 2, 3 and 4? - Show that the polynomial t7 - t takes the value 0 for every value of
t in Zr. (This shows that, in contrast to the complex field, there are
fields in which nonzero polynomials take the value 0 no matter what
value is substituted for the variable.)
Exploration
E.15. Let p be a prime. How many different polynomials of degree n over
Z, are there? Try to find a formula for the number of manic polynomi-
als in Z,[t] of degrees 2, 3, 4 which cannot be expressed as a product of
polynomials of lower degree.
1.8 Problems on Quadratics
- Given that tan A and tan B are the roots of the equation x2+px+q =
0, find the value of
sin2(A + B) + psin(A + B) cos(A + B) + q cos2(A f B).- Find the value of the positive integer n for which the quadratic equa-
tion n.
C( x + i - 1)(x + i) = 10n
i=l
has solutions x = r and x = r + 1 for some number r.
If the coefficient 10 is replaced by an integer p, for which values of p
does a corresponding value of n exist?- Find a necessary and sufficient condition that one root of the quadratic
equation ax2 + bx + c = 0 is the square of the other. - Let p(t) be a manic quadratic polynomial. Show that, for any integer
n, there exists an integer k such that
dn)p(n + 1) = p(k).