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§4. Resultant of Two Polynomials 63

3. Let k[[x]] be the ring of formal series over the field k, that is, expressions of the form
n
a
a(x) = 0≤n n x . Let O(a(x)) = n, where a n = 0and a i = 0 for i < n. Show that
O(ab) = O(a) + O(b), O(a + b) ≥ min{O(a), O(b)}.

Show that the field of fractions k((x)) of k[[x]] is equal to k[[x]][1/x], and that v(a/b) =
O(a) − O(b) is a valuation on k((x)). Moreover, show that the elements c of k((x)) can
i
c
be written in the form c(x) = m≤i i x ,and if c m = 0, then v(c) = m.

4. The discriminant D( f ) of a polynomial f (x) is defined to be the resultant R( f, f ),

where f is the derivative of f . Prove that f in k[x] has a repeated root in an extension
field of k if and only if D( f ) = 0.
2 3
5. Calculate the discriminant D( f ) of the polynomials ax + bx + c and x + px + q.