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1) Let 1 = 8 8i, 2 = 10 + 15i and = 2 3i, and let I = ( ) be the principal ideal of
Z[i] generated by .
(a) Compute the quotient and remainders of the divisions of 1 and 2 by ?
Solution. We divide 1 by :
8 8i
1 5i
(8 8i)(2 + 3i)
40 + 8i
=
1) Prove that if f Z[x] is primitive and g Z[x] divides f in Z[x], then either g or g is
also primitive.
Proof. Let f = g q , where q Q[x]. Write, g = c g0 , q = d q0 , where c, d Q and g0 , q0
are primitive. [So, c and d are the content of g and q respec
Math 456 Midterm I
1) Let R be a ring and I be an ideal of R.
def
(a) Prove that if J is an ideal of R containing I , then J = cfw_a R/I : a J is an ideal
of R/I .
Solution. First observe that if a J , then, there is some b J such that = a, i.e.,
b
a b