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A short class with Sarah and her students. The other ways are under More.
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Good morning everyone! Today, we will talk about finite fields, starting with the field F we constructed with 9 elements. Can anyone remind us what these elements consist of?
They are polynomials of degree 0 and 1 with coefficients from the set {0, 1, 2}!
And we are using ℤ, right? That’s the set of integers modulo 3?
Exactly! Now, can someone explain what closure means in the context of polynomial addition?
Closure means that when we add two polynomials from our set, we get another polynomial that is also in the set.
Great! Yes, adding polynomials within set F keeps us within F. What about multiplication? What challenge did we face with that?
Multiplying polynomials could yield a degree 2 polynomial, which isn't in our set. So we had to adjust the operation.
Correct! That's why we perform multiplication followed by taking modulo the polynomial x² + 1. Let’s summarize: The set F consists of elements structured such that polynomial addition is straightforward while multiplication is modified to ensure closure.
Next, let's discuss the existence of multiplicative inverses in our field. Why is it important for a set to have these inverses?
It's essential because without inverses we can’t satisfy all the field axioms.
If every non-zero element has an inverse, we can do division, like finding 1 divided by any number!
Exactly! For instance, what is the multiplicative inverse of 1 in our field?
It's 1 itself because 1 times 1 is still 1!
And what about the number 2?
The inverse of 2 is also 2 because 2 times 2 equals 4, and 4 modulo 3 is 1, which is the identity element!
Well explained! This process of ensuring each non-zero element has an inverse confirms that our field structure holds. Let's summarize this point: Each non-zero element of the field F has a multiplicative inverse under the defined operations.
Now, let’s talk about the characteristic of our field. What do we mean by the characteristic of a field?
It's the smallest positive integer m such that adding 1 to itself m times gives us 0!
I remember it also relates to the structure of the cyclic group generated by the element 1!
Correct! In our field of 9 elements, what is the characteristic?
It's 3 because if we add the multiplicative identity 1 three times, we reach 3, which modulo 3 is 0.
Right! So how about when we have fields that are not finite? What about their characteristics?
For infinite fields, the characteristic isn’t well-defined. We only concern ourselves with finite fields typically.
That’s correct! The characteristic of finite fields will always be a prime number, ensuring no contradictions in their structure. To summarize, we've defined the characteristic of the field and observed the specific example from our finite field of 9 elements.
A mathematical structure where addition and multiplication operations are well-defined and fulfill field axioms.
Defined as the smallest positive integer such that 'm times 1' equals 0.
Essential for modifying operations to keep closure properties in finite fields.
In fields finite, numbers so bright, 9 together learning at night.
Once upon a time, there were 9 clever polynomials living in a field. They learned how to add and multiply but found out they needed a special rule to stay together under a polynomial called x² + 1.
Remember C for Closure, I for Inverse, and P for Prime when thinking about fields.
FIP: Finite, Inverse, Prime - key features of our field.
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What is a finite field?
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What is the characteristic of a field?
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Give an example of an irreducible polynomial.
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A field with a finite number of elements.
A polynomial that cannot be factored into polynomials of lower degrees over a given field.
The smallest positive integer m such that adding the multiplicative identity 1 to itself m times yields the additive identity 0.
A group that can be generated by a single element.
The property that ensures operations on elements result in other elements within the same set.
This section covers the construction of a finite field with 9 elements by defining polynomial operations over a specified set.
The section discusses how to construct a finite field using 9 polynomials of degree 0 and 1 with coefficients from the set of integers modulo 3. It explains the necessary modifications to operations to ensure closure, the existence of multiplicative inverses, and introduces the concept of the field's characteristic.
In this section, we explore the construction of a finite field, denoted as F, consisting of 9 elements represented by polynomials of degree 0 and 1 with coefficients from the set of integers modulo 3, ℤ = {0, 1, 2}. The operations of addition and multiplication of these polynomials are defined such that they observe closure properties when modified appropriately.
To construct this finite field effectively, we define polynomial addition in a straightforward manner. However, upon performing polynomial multiplication, we noticed that closure is violated; for example, multiplying (x + 2) and (2x + 1) yields a polynomial of degree 2, which does not belong to our set F. Therefore, we modify the multiplication operation by taking the result modulo an irreducible polynomial, specifically x² + 1, to maintain closure.
We subsequently confirm that every non-zero element in the field has a multiplicative inverse, making sure that the set adheres to the field axioms. An interesting property derived is that the non-zero elements of F can be generated using the powers of a specific element, 2x + 1. This establishes that the structure formed is cyclic and demonstrates the concept of the field's characteristic, which is defined as the smallest positive integer m such that adding the multiplicative identity element 1 to itself m times yields the additive identity 0.