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

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.