1.4. Span of the Field

Next let me define what I call as the span of the field. So, a collection of k elements, so, here is your field F which is finite and which has some number of elements. So, if I focus on a collection of values which are called as f1, f2, ..., fk, I will call the collection of these elements as the span of the field if the following hold. You take any element x from the field that can be expressed as a linear combination of the elements from your collection f1 to fk, where the linear combiners are from set 0 to p – 1.

The span of a field refers to a set of elements from which every other element in the field can be created through addition and multiplication. If you have k elements in a finite field F, and if you can create any element x from the field using these k elements with certain coefficients (linear combiners), then these k elements span the field.

Think of a classroom where each student represents a unique element. If a team (the span) can be formed with just a few students (the k elements), and this team can involve roles or assignments that replicate every type of student in the classroom, then the chosen students span the class. Just as not every student needs to be included in the team to perform all roles, not all elements of the field need to be used to span the entire field.

It is easy to see that a trivial span of the field is the entire field itself. You take any element x from the field that can be always represented as 1 times x + all other elements from the field being multiplied with 0.

The entire field can be considered a trivial span because every element can be represented as itself multiplied by 1 plus the sum of 0 times all other elements (which equals 0). This means that the whole field is a valid spanning set because we can express any element in the field trivially using itself.

Imagine a toolbox that contains all the possible tools. If you need to build anything, you can always use tools from this complete toolbox. Here, the toolbox itself represents the trivial span of all the tools, just as the entire field represents the trivial span of its elements.

Now let me next define what we call as the minimal spanning set of the field. So the minimal spanning set of the field is the collection of elements from the field which is minimal in the sense that you cannot remove any element from this collection.

A minimal spanning set contains the smallest number of elements possible from the field such that they still have the property of spanning the entire field. If you remove any element from this set, it will no longer be able to span the entire field, which defines its minimality.

Consider a recipe for making a cake. The minimal spanning set would be the essential ingredients needed to make the cake. If you try to make the cake without any of these minimal ingredients, like flour or eggs, the cake simply won't turn out. Just like these ingredients, the minimal spanning set has all the necessary components to recreate every element in the field.

Now, what I am going to define is the following: I am going to 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.

A mapping from ℤ^r (the set of r-tuples of integers where each integer comes from a set of p values) to the field F allows us to relate integer tuples with elements in our field. This mapping will help demonstrate relationships between different mathematical structures and attributes of the field.

Think of ℤ^r as representing coordinates in a multi-dimensional space. Each tuple can represent a specific point in space, akin to a point that can be expressed in a field through different elements combined together, showing how various elements relate to create something meaningful.

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