Polynomials and function space over fields

In summary, the function E from F[x] to F^F is 1-to-1 if and only if F is infinite, and onto if and only if F is finite. This can be proven by showing that if F is finite, there exists a non-zero polynomial in F[x] that maps to the zero function in F^F, and if F is infinite, every non-zero polynomial in F[x] maps to a non-zero function in F^F. This can also be shown using the Frobenius method and the fact that a finite field has a characteristic. Additionally, if F is a finite field and the degree of a polynomial is greater than the order of F, then the polynomial will map to the zero function
  • #1
kritimehrotra
5
0
Hi,

Can someone explain why the following is true? It seems to be an "accepted fact" everywhere I search, and I can't tell why.

Let F be a field. Let E be the function from F[x] to F^F, where F[x] is the set of all polynomials over F, and F^F is the set of all functions from F to F.
Then E is 1-to-1 if and only if F is infinite, and E is onto if and only if F is finite.

Why is this true?

For the first part, I can see that 1-to-1 would mean that the kernel of E is 0. So if F is finite, there should be some non-zero polynomial in F[x] which maps to the zero function in F^F.

For the second part, I can see that if F is finite, then F^F is finite. F[x] is of course infinite, and so it makes sense that E is onto, but can someone give a more conceptual reason why? Or is that the essence of the reasoning?

Thank you!

Kriti
 
Physics news on Phys.org
  • #2
What do you mean by 'the' function? You simply mean the inclusion right? Well, for the first case, if I take a polynomial and consider it as a function, how can it be the zero function (i.e. send every element of F to zero)? Only if it was the zero polynomial.

The second is also easy. Forget fields. Suppose I tell you, over the reals, that f(0)=1, f(1)=10 and f(3)=4, write down a polynomial through the points (0,1), (1,10) amd (3,4), you'd be able to do that easily. Well, the finite field question you ask is no harder.
 
  • #3
Yes, by "the" function, I meant the natural function mapping each polynomial in F[x] to it's associated function in F^F.

For the first case, what you mentioned is true. I had figured that much myself, but that proves that F is infinite => ker E = 0 <=> E is 1-to-1. How do we go in the opposite direction? I.e., why is it that ker E = 0 => F is infinite?

Similarly, for the second part, what you said was the basic concept I understood. But why does this require a finite field then? Given any function in F^F, couldn't I find some polynomial that could interpolate it?
 
  • #4
If a field is finite it has a characteristic. It is easy to show that (x+1)^p=x^p+1 when p is the characteristic of a finite field. This is more than enough of a prod to help you with the problems you're having. If that isn't enough, then does the word Frobenious help?
 
  • #5
Here is a little something to add to what matt-grime was lecturing about.

Theorem: Let [tex]|F|= \infty[/tex] and [tex]f(x)[/tex] be a polynomial if [tex]f(\alpha)=0[/tex] for every [tex]\alpha \in F[/tex] then [tex]f(x)=0[/tex].

Proof: Let [tex]\deg f(x) = n[/tex] (assuming [tex]f(x)\not = 0[/tex]) then [tex]f(x)[/tex] can have at most [tex]n[/tex] zeros. But it clearly does not for any [tex]\alpha \in F[/tex] is a zero. So [tex]f(x)[/tex] must be the trivial polynomial.

Note, if [tex]F[/tex] is finite and [tex]\deg f(x) > |F|[/tex] then the same conclusion can be drawn. But it need not to be always true. Consider [tex]F[/tex] a field of order 3. And [tex]f(x)=x^3-x[/tex] it maps all elements into zero.
 
  • #6
the proof for surjectivity matt gave also proves non injectivity in the finite case, [besides that it is obvious, since the domain is infinite and the target is finite].
 

1. What is a polynomial?

A polynomial is a mathematical expression that consists of variables and coefficients, combined using operations such as addition, subtraction, and multiplication. It can have one or more terms, and the variables are raised to non-negative integer powers. Examples of polynomials are 2x^2 + 5x + 3 and 4x^3y^2 + 2xy + 1.

2. What is the degree of a polynomial?

The degree of a polynomial is the highest power of the variable in the expression. For example, the polynomial 3x^2 + 5x + 1 has a degree of 2 because the highest power of x is 2. The degree of a polynomial is important because it determines the complexity and behavior of the polynomial.

3. What is a function space?

A function space is a set of functions that share a common property or structure. In the context of polynomials, a function space over a field is a set of polynomials that can be added, subtracted, and multiplied by scalars from that field. The function space over a field is denoted by F[x], where F is the field and x is the variable.

4. What is the significance of function space over fields in mathematics?

Function space over fields is important in mathematics because it provides a framework for studying and analyzing polynomials. It allows for the manipulation and transformation of polynomials using algebraic operations, making it a powerful tool in many areas of mathematics, including algebra, geometry, and calculus.

5. How are polynomials and function space over fields used in real-world applications?

Polynomials and function space over fields have many practical applications in fields such as engineering, economics, and computer science. They are used to model and solve problems involving data analysis, optimization, and control systems. For example, in engineering, polynomials are used to represent and analyze physical systems, and in economics, they are used to model financial markets and make predictions.

Similar threads

Replies
2
Views
936
Replies
5
Views
488
  • Linear and Abstract Algebra
Replies
7
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
799
Replies
6
Views
2K
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
5
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
6
Views
3K
  • Linear and Abstract Algebra
Replies
1
Views
1K
Back
Top