1.6. Mapping from ℤ^r to Field F

Now let me define a mapping g from the ℤ^r to the field F. Now, what is the ℤ^r? So as per the definition of Cartesian product, ℤ^r is nothing but the Cartesian product of ℤ, which itself r times.

In this context, ℤ^r represents a set consisting of tuples made up of r elements, where each element is drawn from the set of integers ℤ. For example, if r = 2, we are dealing with pairs of integer values such as (x, y). This is analogous to points in a 2D coordinate system where each coordinate can take any integer value.

Think of ℤ^r as a multi-dimensional grid where each point is defined by r coordinates. Just like how you find a location on a map using two coordinates (latitude and longitude), here you use r coordinates to define a point in an abstract space.

Now how exactly this mapping g is defined? So if you want to map an r tuple as per the mapping g then what basically you have to do is the following, you have to take a linear combination of the elements in your minimal spanning set as per the linear combiners in your r tuple.

The mapping g takes an r-tuple from ℤ^r and produces an element in the finite field F by calculating a linear combination of the elements in the minimal spanning set. A linear combination means you are combining variables using coefficients (which are elements from the r-tuple) and adding them together. This process ensures that every unique r-tuple corresponds to a unique element in the field F.

Imagine a recipe where each ingredient represents an element from the minimal spanning set. The amount of each ingredient you use corresponds to the coefficients from your r-tuple. So if you have a tuple (2, 1) and the spanning elements are flour and sugar, you would use 2 cups of flour and 1 cup of sugar to create a unique dish (element in F).

Now I am going to make certain claims about this function g. I am going to prove that this function g is a bijection and if it is a bijection then as per the rules of cardinality it shows that the cardinality of F is same as the cardinality of ℤ^r.

A bijection means that there is a one-to-one correspondence between elements in two sets, here between ℤ^r and F. If we can show that for every element in ℤ^r, there is a unique corresponding element in F and vice versa, we can conclude that both sets have the same number of elements, or cardinality. This relationship is essential in mathematics because it helps in understanding the structure of finite fields.

Think of a bijection as a matching game where every player (element from ℤ^r) pairs with one distinct prize (element from F). If every player can be matched with a unique prize without anyone left unpaired, it confirms that the number of players and prizes are equal.

Well, proving that g is a surjection is trivial. That comes from the definition of your spanning set. Since as per my definition, the collection of f to f is a spanning set. That means you give me any element x it will have a pre-image.

Surjectivity means that every element in the field F has at least one pre-image in ℤ^r. Since we defined the minimal spanning set to fully span the field, any element x you pick from F can be constructed using a linear combination from the set of spanning elements using coefficients from ℤ^r, thus confirming surjectivity.

Consider a library where every book represents an element from F. As long as every book can be created using available genres (the spanning set), you have confirmed that every type of book (element) can indeed be created from the genres (mapping from ℤ^r).

Now, I want to prove that this function g is also an injective function and that I will prove by contradiction.

Injectivity means no two different elements in ℤ^r can map to the same element in F. To prove it by contradiction, we assume two different r-tuples map to the same element in F, which would imply that we could express one as a linear combination of the other. This contradicts the minimal spanning set definition, thus proving injectivity.

Imagine you have two different keys that somehow fit into the same lock (mapping to the same element). If both keys work the same way, it means they are essentially interchangeable, which contradicts the uniqueness of each key. By ruling this out, we confirm that each r-tuple must correlate to a unique lock (element in F).

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