19.2.7. Definition of a Field

A field is an algebraic structure; it is a set of values and there are 2 operations; 2 abstract operations plus and dot, which are defined over the elements of this set F and we will say that all together this collection F along with the operation + and ∙ is a field if the field axioms are satisfied.

A field consists of a set of elements along with two operations: addition and multiplication. These operations must follow certain rules, known as 'field axioms.' If these axioms are all met, we can thoughtfully apply concepts like addition and multiplication within this structure.

Think of a field as a team of players (elements) who can play two types of games (operations: addition and multiplication). For the team to function well (satisfy axioms), all players must understand the game rules (field axioms) perfectly.

The axiom number 1 is that the set F along with the plus operation should constitute an abelian group.

For a field, the addition operation must create an abelian group, meaning it must satisfy conditions like closure (adding two elements gives another element in the set), associativity (the way we group elements doesn't change the result), the existence of an additive identity (a zero element), and the presence of additive inverses (every element has a counterpart that sums to zero).

Imagine a classroom where students (elements) are paired to create teams (sums). Each team should always have members from the classroom and should work well together no matter how they are grouped. Plus, every group can team up with another member to balance out during games (inverses).

The second property that we demand here is the following if I exclude the additive inverse and see I have written down here 0 in quote and unquote - it is not integer 0 it is just a representation for denoting the identity element with respect to the plus operation.

This axiom states that when we exclude the zero element, the remaining elements must also form an abelian group under multiplication. This includes closure, associativity, the existence of an identity element, and inverses. Thus, every non-zero number in a field has a multiplicative inverse, ensuring that division doesn't lead us into a dead end (such as dividing by zero).

Consider a magic show where every magician (non-zero element) can perform tricks (operations) and has a magic prop (inverse) that on combining gives an empty stage (the identity). However, the 'no magic' prop, represented by zero, can’t perform magic or have an inverse.

The third property the third field axiom that needs to be satisfied is that your dot should be distributive over plus.

The distributive property ensures that multiplying a number by a sum is the same as multiplying each addend separately and then summing the results. This must hold true in a field context. The operation should look the same whether we apply multiplication to the entire sum first or distribute it across the terms.

Think of a store where discounts (operations) can be applied to an entire bill (sum) or item by item. Whether you find the total discount first or apply it piece by piece before totaling, the end amount remains the same, reflecting the distributive property’s reliability.

You can easily identify that your field is a special type of a ring, where every non-zero element is invertible with respect to the multiplication operation.

While all fields can be classified as rings, not all rings are fields. Fields demand that every non-zero element must have a multiplicative inverse, meaning division is always possible, while rings do not enforce this. This distinction is crucial for various mathematical applications.

Imagine a club (field) where every member (non-zero element) can swap roles (inverses) with another; however, in a general group (ring), some members might not have a partner, making swaps impossible.

Now, it is easy to verify that if I consider a prime modulus then the set of integers, 0 to p - 1, which is nothing but the set ℤ along with the operation addition modulo p and multiplication modulo p satisfies the field axioms.

When using a prime number as a modulus, the integers from 0 to p-1 form a field. This is because all non-zero integers in this set have a multiplicative inverse, satisfying all the field axioms. They can be added and multiplied without exception, maintaining closure, associativity, and inverses.

Imagine a clock where the hour hand moves from 1 to 12 (modulus p). Every hour corresponds to a unique number, and when adding hours (operations), every time remains on the clock—eventually returning to start (closure) but can also 'turn back' (inverse operations, like subtracting hours).

If x ∙ y = 0, then we can safely conclude that it is either the case that your element x is 0 or the element y is 0 that means it will never happen that you take 2 non-zero elements and if you perform the dot operation you get a 0 element that would not happen in a field.

In a field, if the product of two elements is zero, at least one of the elements must be zero. This property highlights the uniqueness of fields versus rings, where this relationship may not hold. Hence, a zero product implies a zero factor.

Imagine a race where two cars (non-zero elements) cannot finish at the exact same time (zero) unless one breaks down (becomes zero). Thus, one must have stopped for it to be a tie, emphasizing that only one car being able to finish is unattainable unless one ceases to exist.

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