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Simple Question on Polynomial Rings

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
When we write [TEX] F[x_1, x_2, ... ... , x_n] [/TEX] 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 [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].]

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]
 
Last edited:

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
Re: Simple question on polynomial ringsWhen we write [TEX] F[x_1, x_2, ... ... , x_n] [/TEX] where F

When we write [TEX] F[x_1, x_2, ... ... , x_n] [/TEX] 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 [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].]

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]
Hey Peter!

I am pretty sure that the former is the case.

Lets take a very simple non-polynomial ring example. When we write $\mathbb R$ we mean the set of all reals, not some specific type of them, like say irrationals or something. There is no reason that mathematicians would choose to use a different and quite ambiguous convention for more complicated structures. :)
 

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
Re: Simple question on polynomial ringsWhen we write [TEX] F[x_1, x_2, ... ... , x_n] [/TEX] where F

Hey Peter!

I am pretty sure that the former is the case.

Lets take a very simple non-polynomial ring example. When we write $\mathbb R$ we mean the set of all reals, not some specific type of them, like say irrationals or something. There is no reason that mathematicians would choose to use a different and quite ambiguous convention for more complicated structures. :)
Thanks caffeinemachine,

You write "There is no reason that mathematicians would choose to use a different and quite ambiguous convention for more complicated structures."

I was more thinking that maybe [TEX] F[x_1, x_2, ... ... , x_n] [/TEX]would stand for a set of possible structures in the same way that when we say, a ring R., it can stand for many structures ... in the same way, I was thinking that maybe [TEX] F[x_1, x_2, ... ... , x_n] [/TEX] could stand for a number of different polynomial rings.

Mind you, I think you are correct anyway :)

Peter
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
Re: Simple question on polynomial ringsWhen we write [TEX] F[x_1, x_2, ... ... , x_n] [/TEX] where F

I was more thinking that maybe [TEX] F[x_1, x_2, ... ... , x_n] [/TEX]would stand for a set of possible structures in the same way that when we say, a ring R., it can stand for many structures ... in the same way, I was thinking that maybe [TEX] F[x_1, x_2, ... ... , x_n] [/TEX] could stand for a number of different polynomial rings.
I don't quite understand you here. Can you please elaborate?
 

hmmm16

Member
Feb 25, 2012
31
Re: Simple question on polynomial ringsWhen we write [TEX] F[x_1, x_2, ... ... , x_n] [/TEX] where F

Thanks caffeinemachine,

You write "There is no reason that mathematicians would choose to use a different and quite ambiguous convention for more complicated structures."

I was more thinking that maybe [TEX] F[x_1, x_2, ... ... , x_n] [/TEX]would stand for a set of possible structures in the same way that when we say, a ring R., it can stand for many structures ... in the same way, I was thinking that maybe [TEX] F[x_1, x_2, ... ... , x_n] [/TEX] could stand for a number of different polynomial rings.

Mind you, I think you are correct anyway :)

Peter
It does stand for a number of different structures in the same way that $R$ stands for different structures but that is because the $F$ can represent different fields.

So for example the polynomial ring $\mathbb{Q}[x_1,.....,x_n]$ has co-efficients from the rationals and would be analogous to the ring $\mathbb{Q}$

And $F[x_1,.....,x_n]$ has co-efficients from the field $F$ whatever that may be in the same way that $R$ has elements from $R$ whatever that may be.

However once we specify this field it does not then change