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About LinkBacksLet R be a commutative ring. Let A0 be an ideal in R[x] such that
the lowest degree of a nonzero polynomial in A is n1 and such that A
contains a monic polynomial of degree n. Prove that A is a principal ideal
in R[x].
letOriginally Posted by mathman88
Let R be a commutative ring. Let A
the lowest degree of a nonzero polynomial in A is n
contains a monic polynomial of degree n. Prove that A is a principal ideal
in R[x].be a monic polynomial of degree
obviously
now using induction over degrees of elements of
we show that every elment of
which will
prove thatso let
then
is a polynomial of degree at most
and thus by the minimality of
we have
-a_0f(x)=0.)
thusthis proves the base of the induction. now let
and suppose every polynomial in
is in
).)
letthen
and
thus by induction hypothesis
for some
then
=(a_0x^{m-n}+u(x))f(x) \in(f(x)).)
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