Proving V+J=k[x] using induction for k with polynomial ring in n variables

  • Thread starter grimster
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In summary, the conversation discusses using induction to prove that the sum of two sets, V and J, equals the polynomial ring in n variables over a field with q elements. The method involves using the notion of L_j, the sum of degrees of variables greater than q in a monomial, and showing that this holds for a given polynomial f in k[x]. The individual must then reduce all monomials in f with exponents greater than q in order to prove the statement.
  • #1
grimster
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J=sum from i=1->n of <X_i^q - X_i>
V={polynomials f with deg in X_i < q)

k is a field with q elements. k[X] is the polynomial ring in "n" variables".

i am supposed to prove that V+J=k[x].

i was told that this could be done with induction for the number "k", by using the following notion.given a polynomial f E k[x]:

k0=the sum of L_j. j=1..."number of monomials".
L_j is the sum of of the "degrees" of the "indeterminates/variables" which are >q in monomial "j".

so L_j=sum of all s_i, where s_i > q. s_i are the exponents of the monomial which are >q.

so somehow i have to show that this is true for <k0 and then it is true for k0. or something like that?

does anyone know what I'm saying here? it might be a little confusing. just ask if there's anything you don't understand.
 
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  • #2
[itex]$ k0=\sum\limits_{j\in \left\{ 1,...,\text{number of monomials}\right\} }L_{j}$[/itex]
 
  • #3
i was thinking that if given a polynomial f in k[x].

then if i reduce all monomials in f such that all exponents are < q. then the reduced f will be in V, not?
 

1. What is the significance of proving V+J=k[x]?

Proving V+J=k[x] has several implications in mathematics and science. It helps establish a relationship between two sets of numbers, V and J, and shows that their sum is equal to another set of numbers, k[x]. This can be useful in solving equations, analyzing data, and understanding mathematical concepts.

2. How is the proof for V+J=k[x] conducted?

The proof for V+J=k[x] is typically done using mathematical principles and equations. It involves manipulating the equations to show that the sum of V and J is equal to k[x] by using properties such as the distributive, commutative, and associative properties of addition.

3. Can you provide an example of a proof for V+J=k[x]?

One example of a proof for V+J=k[x] could be as follows: Let V={1,2,3} and J={4,5,6}. We can write the sum of V and J as V+J={1,2,3,4,5,6}. Now, let k[x]={5,7,9}. Using the commutative property, we can rearrange the elements in V+J to get V+J={1,4,2,5,3,6}. This is equivalent to k[x], and thus we have proven that V+J=k[x].

4. What are the main challenges in proving V+J=k[x]?

One of the main challenges in proving V+J=k[x] is understanding the properties and rules of addition and how they can be applied to manipulate equations. Another challenge is finding the right approach to prove the equation, as there can be multiple ways to do so. It also requires a strong understanding of algebra and mathematical concepts.

5. How can the proof for V+J=k[x] be applied in real-life situations?

The proof for V+J=k[x] can be applied in various real-life situations. For example, it can be used to solve equations in physics and engineering, analyze data in statistics, and understand relationships between different variables in economics and social sciences. It can also be used to develop new mathematical theories and principles.

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