An ideal I of a commutative ring R is said to be principal if it is generated by a single element, that is, if I is of the form
{ra|r element of R} for some fixed a element of R. Notice that Corollary 6.6 (below) shows that ideals of F[x] of the form I_F,a are principal. Now prove that every ideal of F[x], where F is any field, is principal. (Hint: If I = {0}, then we are done - why? Otherwise, there must be some nonzero element in I of least degree, call it q. Given an arbitrary element f is a member of I, write f = bq +r (division algorithm). Use the definition of an ideal to show that r must be in I. Conclude from the choice of q and r that r must be zero. This shows that I is a subset of{rq|r is an element of F[x]}. Why does the reverse inclusion also hold?)

Corollary 6.6 - If p(x) is any nonzero element of I_F,a of least degree,then I_F,a = {f(x)p(x) |f(x) is an element F[x]}.

Solution Preview

Let I be an ideal of F[x].

If I = 0, then I = <0> is certainly principal.

Suppose then I =/= {0} and let g(x) be a nonzero element of I of minimal degree. If the degree of g(x) is 0, then ...

... Integral domains and principal integral domains are exemplified in this ...Proof: " ⇒" : Suppose R is a PID, for any a, b ≠ 0 , we consider the ideal I =< a ...

... Using the least integer principle, define r to be the least ... 3. Use Exercise 2 to give a new proof of Theorem ... two conditions is called an integral ideal: (i) If ...

... other words, reason alone cannot provide a proof for this ... on this also - that humans do have ideals for which ... was to strictly adhere to the principle "lying is ...

... probably the most controversial of the great social ideals. ... Why is such equality an ideal, and equality of ... A principal objection to libertarian theory is that ...

... Continue to do this until you reach the ideal state of needing ... fact that Toyota endorses this principle and utilizes this principle, is further proof to the ...

... on the other hand, are guiding ideals of conduct ...Principle 11 A physical therapist shall respect the rights ... by a judge or jury; jury decision Proof need not be ...

... An optimal solution is ideal indeed but a satisficing ... His conditions require no proof and what he has ... to prepare for a test uses the principle of satisficing ...

... pain and physical decline violate ideals of personal ... If the harm principle is interpreted in utilitarian ... the downfall of the rehabilitative ideal43 and ...

... integration, and structural integration serves the ideal of equal ... In fact, this was the principal rationale used by the ... at the University has been proof for me ...

... tied together in some way, some unifying principle, and by ... and ethical relativism, what side is most ideal?/Is there ... this regard isn't like a proof in geometry ...