# Simple question on polynomial rings

• Jun 30th 2013, 02:49 AM
Bernhard
Simple question on polynomial rings
When we write \$\displaystyle F[x_1, x_2, ... ... , x_n] \$ where F is, say, a field, do we necessarily mean the set of all possible polynomials in x_1, x_2, ... ... x_n with coefficients in F? [In this case, essentially all that is required to determine whether a polynomial belongs to \$\displaystyle F[x_1, x_2, ... ... , x_n] \$ is to check that the co-efficients belong to F and the indeterminates only contain \$\displaystyle x_1, x_2, ... ... , x_n \$.]

OR

when e write \$\displaystyle F[x_1, x_2, ... ... , x_n] \$ do we mean to include possible cases such as the set of polynomials with even coefficients - that is we may be talking about the set of polynomials with even co-efficients - so we cannot be sure what ring of polynomials we are talking about when we write \$\displaystyle F[x_1, x_2, ... ... , x_n] \$ until we specify the exact nature of ring of polynomials we are talking about further.

If the latter is the case when given \$\displaystyle F[x_1, x_2, ... ... , x_n] \$ we can not reason about whether particular polynomials belong to \$\displaystyle F[x_1, x_2, ... ... , x_n] \$ until you know the exact nature of the ring \$\displaystyle F[x_1, x_2, ... ... , x_n] \$

I very much suspect that the former is the case but ... ... Can someone please confirm or clarify this.

Peter
• Jul 1st 2013, 02:46 PM
Drexel28
Re: Simple question on polynomial rings
Hello,

I am a little confused as to how you would have this confusion. You are correct about the former--the ring \$\displaystyle F[x_1,\ldots,x_n]\$ literally means the ring of all polynomials in the indeterminates \$\displaystyle x_1,\ldots,x_n\$ with coefficients in \$\displaystyle F\$. Things in math are very rarely wishy-washy enough so that this symbol could be as ambiguous as you claim.

I hope this helps.

PS Evenness makes no sense in the context of fields.