4.3. Solving Special Polynomial Equations 129
We might try to factor it as the product of a quadratic and a cubic:(t2 + ut + v)(t3 - ut2 + wt + z).By comparing coefficients, obtain four equations for U, v, w, z in terms
of a, b, c, d. Eliminate the variables v, w, z and obtain a polynomial
equation for u. What is the degree of this equation? Can the value of
u be found by solving an equation for a power of u whose degree is
less than 5?- If a polynomial over a field does not have a zero in the field, then it
is possible to find a larger field containing the coefficients which also
contains a zero. Let us look at the situation when the polynomial is
quadratic and the coefficient field is Q.
Suppose d # 1 is an integer which is not divisible by any perfect
square except 1. Then 4 is nonrational and so the polynomial t2 - d
has no zero in Q. However, we can extend Q to a larger field by
“adjoining” a. Let
Q(&)={a+b&: a,bEQ}.(a) Verify that fi, 1 + 2fi and (l/3)(7 - 4&) are members of
Q(d), but that 4, 31i3 and i are not.
(b) Is i a member of Q(G)?
(c) Show that Q(i) # C.
(d) Verify that Q(d) is closed under the operations of addition,
subtraction and multiplication.
(e) The surd conjugate of a+b& is defined to be a-b&. Verify that
the product of any number in Q(d) with its surd conjugate is
rational, and deduce that the reciprocal of any element in Q(A)
is also in Q(h).
(f) Show that Q(a) is th e smallest field which contains all of Q
along with the number 4.
(g) Write t2 - d as a product of linear factors over Q(d).
(h) Determine integers b and c such that bc # 0 and t2 + bt + c is
irreducible over Q and reducible over Q(a).- The role played by Q in Exercise 6 can be played by any field F.
Thus, F(a) = {a + b&i : a, b E F}.
(a) Verify that C = R(G).
(b) Let F = Q(fi). Show that F(a) consists of numbers of the
form a + bfi + cd + dfi, where a, b, c, d E Q. Determine
(a+bfi+c&+d&)-‘. (Ob serve that Q(d)(d) = Q(&>(fi>,