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- Jun 22, 2012

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When we write [TEX] F[x_1, x_2, ... ... , x_n] [/TEX] where F is, say, a field, do we

OR

when e write [TEX] F[x_1, x_2, ... ... , x_n] [/TEX] 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 [TEX] F[x_1, x_2, ... ... , x_n] [/TEX] until we specify the exact nature of ring of polynomials we are talking about further.

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

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

Peter

[This is also posted on MHF]

__mean the set of__**necessarily**__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 [TEX] F[x_1, x_2, ... ... , x_n] [/TEX] is to check that the co-efficients belong to F and the indeterminates only contain [TEX] x_1, x_2, ... ... , x_n [/TEX].]__**all**OR

when e write [TEX] F[x_1, x_2, ... ... , x_n] [/TEX] 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 [TEX] F[x_1, x_2, ... ... , x_n] [/TEX] until we specify the exact nature of ring of polynomials we are talking about further.

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

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

Peter

[This is also posted on MHF]

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