24.2. Verification of Field Axioms

It turns out that with respect to the operation of polynomial addition over the set ℤ [x], then it satisfies the closure property namely, you take any 2 polynomials from this collection and add them you will get again a polynomial in the same set F.

The closure property states that when you perform an operation (in this case, addition) on members of a set, the result is also a member of that set. Here, if you take any two polynomials from the set of polynomials with coefficients from ℤ, their sum is also a polynomial with coefficients from ℤ. Thus, the set is closed under polynomial addition.

Think of a box of toys—if you take two toys out from the box and combine them into one new toy, as long as the new toy is also in the box, you still have a complete toy collection. This is like the closure property in addition.

But it turns out that with respect to the operation of polynomial multiplication, the closure property is not satisfied.

In polynomial multiplication, if you multiply two polynomials (for example, (x + 2) and (2x + 1)), the resulting polynomial may increase in degree and can produce a polynomial that is not included in the original set. This violation shows that the set is not closed under multiplication.

If you imagine mixing two colors of paint, sometimes the resulting color doesn't match any of the original paints in your selection. This inaccessibility of results corresponds to not satisfying the closure property.

So, what I am going to do is I am going to define a modified addition and multiplication operation, where I will be doing all the addition and multiplication of the polynomial as I was doing earlier, but my resultant answer will be computed modulo this polynomial (x^2 + 1).

To address the closure issue, we modify how we perform operations on our polynomials by reducing results modulo an irreducible polynomial (in this case, x^2 + 1). This means that if a polynomial's degree exceeds a certain limit, we adjust the polynomial back into the set by using the polynomial division remainder. This way, we maintain closure under our operations.

This is like setting rules in a game; when you reach a certain score, instead of exceeding the limit, you reset to zero or a certain value. The modified operation keeps us within our defined set.

Now, I would be interested to check whether this collection F indeed satisfies the axioms of field as well, with respect to the addition and multiplication operation modulo this irreducible polynomial.

A crucial axiom of a field is that every non-zero element must have a multiplicative inverse. This means for any non-zero polynomial in the set, we should find another polynomial such that their product equals the multiplicative identity (which is 1). This step is necessary to confirm that our collection behaves like a field.

Think of having a set of keys: every key needs a matching lock to be truly functional. Ensuring every non-zero polynomial has an inverse acts like finding the matching lock for each key.

For instance, the multiplicative inverse of 1 is 1 because if you multiply 1 with 1, you get 1 and then if you do a modulo x^2 + 1 you will get 1.

Here, we verify that the element 1 serves as its own multiplicative inverse, reinforcing that it fulfills the requirement that every non-zero element should have an inverse. Similarly, we check other elements in our set, ensuring each has a corresponding inverse.

This is like a see-saw balance: if one side weighs one kg, then adding a one kg weight on the other side keeps the balance. Each element and its inverse keep the overall system in balance.

Now, I want to show you some another interesting property of this field F. So, we have already proved now that this collection F satisfies the field axioms with respect to the addition and multiplication operation modulo this irreducible polynomial.

After proving that our set satisfies the necessary axioms to be a field, we also analyze the characteristic of this field. The characteristic represents the smallest positive integer such that adding the multiplicative identity (1) to itself that many times results in the additive identity (0). Understanding the characteristic helps inform us about the structure and behavior of the field.

If you think of a clock with a certain number of hours, the characteristic can be seen as how many hours it takes before you return to the starting point (0:00). Every full cycle is linked to the characteristic number.

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