Re: Z(R)-Module


"Arturo Magidin" <magidin@xxxxxxxxxxxxxxxxx> wrote in message
news:egm8nn$28a3$1@xxxxxxxxxxxxxxxxxxxxx
In article <452ea4d0$0$172$edfadb0f@xxxxxxxxxxxxxxxxxxxx>,
Frederik <None@none> wrote:
Let R=C[x,sigma] be the ordinary polynomial ring with respect to addition.
Elements are written Sum(a_i x^i) where a_i is in C (the complex numbers).
Multiplication is defined as x a = sigma(a) x

And what is sigma? Complex conjugation? Something else?

yeah. sigma(a) is complex conjugation.

The center of R is Z(R)={ polynomials of the form a0+a2x^2+a4x^4+...where
a2,a4,... are real numbers }

Show that R is a finitely generated Z(R)-module.

Find a finite number of polynomials with complex coefficients, p1,
..., pn, such that every polynomial with complex coefficients can be
written as

q1*p1 + ... + qn*pn

where q1,....,qn are real polynomials in x^2.

HINT: If q1 = 1, and q2=x, the Z(R)-submodule generated by q1 and a2
contains all polynomials with real coefficients. What can you add so
that you have all polynomials with COMPLEX coefficients?

Do you really mean q1=1 and q2=x (and not p1=1 and p2=x )?

R is a Z(R)-module generated by 1,x,i,ix


.

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