Can Polynomial Maps and Ideals Demonstrate a Ring Isomorphism in Finite Fields?

• grimster
In summary, the conversation is about proving the equivalence between the ring of polynomial mappings $Map(n,K)$ and the quotient ring $K[X_{1},..X_{n}]/I$, where $I$ is the ideal generated by the elements $X_{i}^{q}-X_{i},1\leq i\leq n.$ The conversation delves into the definition of polynomial maps and the use of homomorphisms to show the isomorphism between the two rings. The conversation also discusses the steps involved in proving that $I=J$, including reducing exponents and using a corollary from lang(p.177 c.1.8). The conversation concludes with mentioning the need to prove that the
grimster
ok, I've pasting some of the stuff I've done in scientific workplace 3.0. should be easier to read than in plain text. hope some of you can help me... just ask if there is something you don't get.

I am supposed to prove that $Map(n,K)\thickapprox K[X_{1},..X_{n}]/I.$ where I is the ideal generated by the elements $X_{i}^{q}-X_{i},1\leq i\leq n.$ Map(n,K) is the ring of polynomial mappings. here is a link that explains what a polynomial map is:

http://mathworld.wolfram.com/PolynomialMap.html

K is a field of q elements.

have a homomorphism $\phi :(K[X_{1},..X_{n}])\rightarrow GMAP(K^{n},K)$ where GMAP is the group of all mappings from $K^{n}\rightarrow K.$

this is an evaluation homomorphism. $\phi (f):K^{n}\rightarrow K$ and $\phi (f)(a_{1},...a_{n}):=f(a_{1},...,a_{n}).$ $I=\ker \phi .J=<X_{i}^{q}-X_{i}>.$

here is my plan. i want to use the first isomorphism theorem.

if we have a homomorphism $f:G\rightarrow H,$ then we have that $G/\ker f\thickapprox \func{Im}f.$ In my case G is $K[X_{1},..X_{n}]$ and H is $GMap(K^{n},K)$

to do this i have to prove first that I=J. This can be done by showing that $I\subseteq J$ and that $J\subseteq I.$ i must also prove that $\func{Im}f=Map(K^{n},K)=K[X]/I.$

we have that every X$^{q}$ can be replaced by X. given a polynomial $f=\sum_{i_{1},...,i_{n}}a_{i_{n},...,i_{n}}X_{1}^{i_{1}}X_{2}^{i_{2}}...X_{n}^{i_{n}}.$ What i want to do is reduce all parts, so that all exponents are $\leq q.$

given $f=\sum_{i_{1},...,i_{n}}a_{i_{1}},...,_{i_{n}}X_{1}^{i_{1}}X_{2}^{i_{2}}...X_{n}^{i_{n}}\in I.$ if i.e. i$_{1}\geq q,$ then i would i have $f^{|}=f-(X_{1}^{q}-X_{i})\cdot a_{i_{1}},...,_{i_{n}}X_{1}^{i-q}X_{2}^{i_{2}}...X_{n}^{i_{n}}.$ This is done for all i, so that every exponent is $\leq q.$ Then $f^{|}\in I.$ An element in both I and J. The monomial $a_{i_{1}},...,_{i_{n}}X_{1}^{i_{1}}X_{2}^{i_{2}}...X_{n}^{i_{n}}$ ''becomes'' $a_{i_{1}},...,_{i_{n}}X_{1}^{i_{1}-q+1}X_{2}^{i_{2}}...X_{n}^{i_{n}}.$

so $f^{|}\in I.f-f^{|}\in J.$ $\deg f^{|}\prec ($less than) $q.$ then a corrollary from lang(p.177 c.1.8) says:let k be a finite field with q elements. let f be a polynomial in n variables over k such that the degree of f in each variable is less than q. if f induces the zero function on k$^{(n)}$ then f=0.

so considering this, $f\in J.$ Does all of this make any sense or am i waaaay off here? how do i show the oter way around? $J\subseteq I?$

i know it's a lot to read, but bare with me here:

so when one has shown that I=J, i must prove the other part.

$V=\{$polynomials with $\deg x_{i}f\prec q\}.$ a vector space over K. $dim_{k}V=\{$the number of different monomials\}= q$^{n}.$ then $|V|=q^{q^{n}}$.

we have linear mappings $V\rightarrow K[X_{1},...,X_{n}]\rightarrow K[]/I.$ So if i show that ker = 0 and that this is surjective(from V to K[]/I) then we have an isomorphism from V to K[]/I. $\func{Im}\phi =Map.$ So then from the isomorphism theorem, G/$\ker f\thickapprox \func{Im}f.$ That should give $K[X_{1},...,X_{n}]/I\thickapprox map(n,K).$

You can use these tags to generate LaTex: [ itex ] ... [ /itex ]. (Without the excess spaces)

For example, the first LaTeX expression in your post becomes $Map(n,K)\thickapprox K[X_{1},..X_{n}]/I.$

Firstly, I would like to commend you for the thorough and detailed work you have done so far. It is clear that you have a good understanding of the concepts and have put in a lot of effort to prove this isomorphism. However, there are a few areas where I believe you may have gone off track or could improve upon.

Firstly, in your plan to use the first isomorphism theorem, you mention that you need to prove that $I=J$ in order to apply the theorem. However, this is not entirely necessary. The first isomorphism theorem states that if we have a homomorphism $f:G\rightarrow H,$ then $G/\ker f\thickapprox \func{Im}f.$ In your case, you have already defined the homomorphism $\phi: K[X_{1},...,X_{n}]\rightarrow \func{Map}(K^{n},K)$ and have shown that $I=\ker \phi.$ Therefore, you can directly apply the first isomorphism theorem without needing to prove that $I=J.$

Additionally, in your proof that $I\subseteq J,$ you have shown that any polynomial $f\in I$ can be written as $f^{|}\in I.$ However, this is not sufficient to prove that $I\subseteq J.$ In order to show this, you must also show that $f^{|}$ can be written as a linear combination of elements in $J.$ This is necessary because $J$ is defined as the ideal generated by the elements $X_{i}^{q}-X_{i},$ so any element in $J$ must be a linear combination of these elements. Similarly, in your proof that $J\subseteq I,$ you need to show that any element in $J$ can be written as a linear combination of elements in $I.$ This will require some more work, but it is possible to show that $J\subseteq I$ in a similar manner to how you showed $I\subseteq J.$

Finally, in your proof that $\func{Im}\phi= \func{Map}(K^{n},K),$ you mention that you have linear mappings $V\rightarrow K[X_{1},...,X_{n}]\rightarrow K[]/I.$ However, it is not clear how you have defined these linear mappings. It seems that you are trying to use the evaluation homomorphism

1. What is an isomorphism?

An isomorphism is a mathematical concept used to compare two structures or systems. It is a mapping between two objects that preserves their structure and properties.

2. How do you prove two structures are isomorphic?

To prove that two structures are isomorphic, you need to show that there exists a one-to-one mapping between the elements of the two structures that preserves their relationships and operations.

3. What is the significance of proving isomorphism?

Proving isomorphism is important because it allows us to understand the relationship between two seemingly different structures and apply knowledge from one to the other. It also helps us classify and categorize structures based on their similarities and differences.

4. What are some common methods for proving isomorphism?

Some common methods for proving isomorphism include showing that the structures have the same number of elements, demonstrating that the operations and relationships are preserved, and using graph theory or algebraic techniques.

5. Can two structures be isomorphic if they have different elements?

No, two structures must have the same number of elements to be isomorphic. If the number of elements is different, it means that there is no one-to-one mapping between the elements of the two structures, and therefore they cannot be isomorphic.

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