xSYS

Are these examples of Fields ?

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Posted (edited)

Hi,

I'm trying to get some insight on the concept of Fields. Definitions abound on the net, and i think i have a clear picture of them. I'd like to know if my following interrelated ideas are correct:

(1) A polynomial is a field.

(2) A set of polynomials is a field

Now i think (2) creates more possibilities for discovery, and maybe it is even the correct way to go if one talks about Fields, whereas (1) is more restricted in that the coëfficiënt-space and power space are smaller; or 1-dimensional maybe, 

So, if i see a polynomial, i see a Field because the four operations are defined (the "sum" of monomials where then the monomials are a "product"), whereas i'd rather prefer two; addition and multiplication thus excluding the reverse and inverse ops. Am i making a mistake here, as in that there NEED to be four? And if there are only two, then can it be that it is related to algebraïc number theory?, because i seem to remember some books where only addition and multiplication are used in some number theory,    

Is it then reasonable to define:

- A sequence of coëfficiënts, say the triangular numbers.

- A sequence of powers, say the pentagonal numbers.

or

- A sequence of binomial coëfficiënts with elements a and where the result is a sum of these types of terms: (a_i + b_j)x^n

- A mapping between these sequences giving more meaning to the concept of Field

- And a host of infinite possibilities for combination and mapping.

etc...

 

 

Edited by xSYS
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13 minutes ago, xSYS said:

Hi,

I'm trying to get some insight on the concept of Fields. Definitions abound on the net, and i think i have a clear picture of them. I'd like to know if my following interrelated ideas are correct:

(1) A polynomial is a field.

(2) A set of polynomials is a field

Now i think (2) creates more possibilities for discovery, and maybe it is even the correct way to go if one talks about Fields, whereas (1) is more restricted in that the coëfficiënt-space and power space are smaller; or 1-dimensional maybe, 

So, if i see a polynomial, i see a Field because the four operations are defined (the "sum" of monomials where then the monomials are a "product"), whereas i'd rather prefer two; addition and multiplication thus excluding the reverse and inverse ops. Am i making a mistake here, as in that there NEED to be four? And if there are only two, then can it be that it is related to algebraïc number theory?, because i seem to remember some books where only addition and multiplication are used in some number theory,    

Is it then reasonable to define:

- A sequence of coëfficiënts, say the triangular numbers.

- A sequence of powers, say the pentagonal numbers.

or

- A sequence of binomial coëfficiënts with elements a and where the result is a sum of these types of terms: (a_i + b_j)x^n

- A mapping between these sequences giving more meaning to the concept of Field

- And a host of infinite possibilities for combination and mapping.

etc...

 

 

(1) No a field can't be a set with a single member. The coefficients of a polynomial are drawn form a field.

(2) Yes some sets of polynomials can form a field.

Not sure what the rest of your post is about though.

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Not sure what the rest of your post is about though.

No maybe because my initial assumption (1) is wrong and therefore things that follow can't possibly make any sense. But if you look at (2) then maybe it makes sense if you consider a matrix of coëfficiënts which then are defined as elements of the polynomials forming a Field. And then define mappings between them; an example would be somthing similar as the proces of say deriving a determinant.

 

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Posted (edited)

Still not sure what you are trying to do.

(1) Fails the identity axiom: "there exist two different elements 0 and 1 in F such that a + 0 = a and a · 1 = a. "

You can use either the 2 binary or 4 binary operations definition, they are equivalent since the second 2 reverse or undo the first two operations.

 

The simplest field is the set {0, 1}

 

Edited by studiot
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Still not sure what you are trying to do.

(1) Fails the identity axiom: "there exist two different elements 0 and 1 in F such that a + 0 = a and a · 1 = a. "

You can use either the 2 binary or 4 binary operations definition, they are equivalent since the second 2 reverse or undo the first two operations.

Ok, but what if you consider a polynomial as a set of monomials (which are added) and the monomials as a 2-element set consisting of a coëfficiënt and a variable which are multiplied. Then the identity axiom holds doesn't it ? "a + 0 = a" for the added monomials and  "a x 1 = a" for the elements of the monomials.

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So it's not a field.

To get a field whose elements are polynomials, or even more complicated, rational functions (i.e. the ratio of two polynomials) you have to allow for powers higher than appear in the original polynomials.

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Posted (edited)

 

Quote

So it's not a field.

We are on two different levels here. What about a Field of monomials ? I am talking about these, about (1).

Quote

To get a field whose elements are polynomials, or even more complicated, rational functions (i.e. the ratio of two polynomials) you have to allow for powers higher than appear in the original polynomials

This is (2). And i can understand this. You actually answered my question. The powers which are higher than the previous ones does not have to form the more natural sequence of 1, 2, 3, 4 but can actually be say the even and odd numbers or  a sequence of figurate numbers as i mentioned. So then it is permitted to rationalize polynomials, but they can also be multiplied, added and subtracted. Your "rationalization" statement is therefore arbitrary, as i am talking about the coëfficiënts OR the powers considered IN the "subfields" or something like that, if it exists. If it does not, then we'll better stay on the level of discussion of (1) , added then; where you are then right in incorporating the "powers" in your argument. 

Edited by xSYS
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Posted (edited)

The 1 and 0 and the letters a and b are just 'dummy' elements or placeholders.

 

There is only one 'field' with 2 elements.

Whatever those two elements are, they must be the same type of object and they must conform to the operation tables as shown in the link.

https://en.wikipedia.org/wiki/GF(2)

Interesting as this is, I'm sorry but it's now half past one in the morning so I will have to adjourn till the morrow.

Edited by studiot
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Posted (edited)

Ok,

sorry i'm trying to make sense out of all this.

to reïterate (1):

A polynomial is a Field.    

So then if it is a field, then it must have some elements with at least two operations defined on them. These have to satisfy the identity axiom. For this to be true, the elements have to be added AND multiplied. Now, if you consider A polynomial is a Field of monomials, then the monomials are elements of the polynomial. And a polynomial consists of added monomials. This satisfies the additive identity. But when you consider the monomials, then these have the multiplicative property implicitly defined on them. but when you consider them on the level of a Field, it doesn't make any difference where the multiplication is defined. In this case, the monomials are a magma, which can be considered as a subfield with only one operation, namely multiplication, which is an element of the Field. This satisfies the identity axiom of multiplication IN the Field in the form of a magma. So a magma is a subfield, and a subfield belongs to a more general Field, the field of the single polynomial...

or am i still missing something here ?.isn't a magma a kind of subfield ? Do i have to include the operations themselves as elements for this to work, thus to define a subfield as a field with only one operation ?

 

but my gut says i'm wrong. I stumbled upon an inconsistency when i tried to define a monomial itself as a field. So i had to incorporate a magma.

 

but mybe i'm making things to complicated. I don't know, when i look at a polynomial i see addition of monomials and multiplication of the elements of a monomial...theres the culprit. Maybe i need to find a way to map a field to a magma But then the result wouldn't be a field anymore ithink..who knows what it is...

 

 

 

Edited by xSYS
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