Question

Is the ring of polynomials with real coefficients a field?

Answer 1

so ring axioms + abelian group under multiplication?

Answer 2

I honestly don't know, I just have a specific doubt because I was told no because of this counter example of an inverse isn't in the field:

\[\frac{1}{1+x}\]

However I believe it is because:

\[\frac{1}{1+x}=1-x+x^2-x^3+\cdots\]

So as far as I'm concerned this is in the field.

What makes a difference if it's infinite, just because this particular string isn't by convention omitted since they're multiplying zeroes?

\[1+0x+0x^2+0x^3+\cdots\]

That infinite polynomial is in the same space and are on equal footing it seems.

Answer 3

How do you get used to this complicated mathematical vocabulary? I thought you had a grudge against mathematics for that very reason.

Answer 4

parth kohli can u help me with a science question

Answer 5

sorry to interupt but its urgent

Answer 6

I don't wanna but I think I figured out how to create a calculus with its own fundamental theorem of calculus on any any space that can be tiled or any connected graph with vertices @ParthKohli lol

Answer 7

how did you get graph theory in here
that almost sounds made up to a beginner like me

Answer 8

do polynomials have to be continuous everywhere? if the answer is yes, then even though 1/(1+x) can be written as a never ending polynomial it still doesn't exist at x=-1.

Answer 9

I mean so really it isn't a polynomial

Answer 10

I don't know what a polynomial is I guess, it just seems to me that it has to be written as something of this form:

\[\sum_{n=0}^\infty a_nx^n\]

But I guess I'm not too worried about this, it just seems like any definition is going to be arbitrary anyways so I think I'll just accept it and move on.

Answer 11

http://math.stackexchange.com/questions/2514/why-cant-the-polynomial-ring-be-a-field
here is another discussion on it

Answer 12

I didn't read all the answers because some of them had fancy stuff in it and I didn't really want to look at fancy stuff.

Answer 13

That's funny... http://math.stackexchange.com/a/2517/207119

Answer 14

do you see my name there

Answer 15

the weird thing is parth edited that 3 years after if I'm reading the dates correctly.

Answer 16

3 years after the question was asked

Answer 17

hey don't make fun of me
i just desperately wanted a badge for editing lol

Answer 18

and it's been three years ever since

Answer 19

haha

Answer 20

you know the definitions I have looked at define
define a polynomial as

\[\sum_{k=0}^n a_kx^k\]

they don't really say if n has to be finite or not

Answer 21

in some contexts using\[\frac{1}{1-x}=1+x+x^2 + \cdots\]helps such as in generating functions

Answer 22

actually it might
it does say where n is a non-negative integer
and infinity isn't even a number

Answer 23

Still, I feel like "finite polynomials" are really just infinite polynomials where you have omitted all the terms multiplying zero while the geometric series are essentially the same thing just with all ones as coefficients. Oh well, I'm not too worried about this, it's just definition that really has no consequence as far as I can tell.

Answer 24

I think your question is interesting because it makes me think that the definition of a polynomial either hasn't been defined very well or you have found a hole in someone's logic or whatever.
I think your question should be upvoted way more than the one on the mathexhange thread because you were able to show 1/(1-x) is an infinite polynomial whatever polynomial really means.

Answer 25

I mean to say you gave an example of there being a polynomial multiplicaitive inverse for 1-x

Answer 26

Yeah, like I often believe that since there are infinitely many primes, then really we're only looking at numbers we can count up to with mostly 0 exponents. Why not just extend to infinite numbers with prime factorizations that have something like this:

\[n = \prod_{n=1}^\infty p_n^{1}\]
We are already fine with this:
\[1 = \prod_{n=1}^\infty p_n^{0}\]

It's just somehow because no one has a notation to write down large numbers we sorta just say "whatever they don't exist" or something? Meh, seems sorta arbitrary if you ask me, anyways.

Answer 27

https://www.reddit.com/r/askscience/comments/39k61d/since_taylor_polynomials_have_an_infinite_number/
W.T. Jones (whoever this is) believes a polynomial to only have finite terms
"Taylor polynomial is a truncated Taylor series"