Sunday, November 6, 2016

abstract algebra - Constructing finite fields of order $8$ and $27$ or any non-prime

I want to construct a field with $8$ elements and a field with $27$ elements for an ungraded exercise.







For $\bf 8$ elements: So we can't just have $\Bbb Z/8\Bbb Z$ since this is not even an integral domain. But rather we can construct $\Bbb F_2 \oplus \Bbb F_2 \oplus\Bbb F_2 \oplus \Bbb F_2 = \{0,1,\alpha,\alpha+1,\beta,\beta+1,\gamma,\gamma+1\}$.



This line of thinking seems to break from what I tried. Is there a better way to construct these things?



I saw this answer: Construct a finite field of order 27



We pick a polynomial irreducible polynomial and take the quotient of $\Bbb Z_3[x]$ but this wasn't helpful in me understanding the general ideal/method.

No comments:

Post a Comment

analysis - Injection, making bijection

I have injection $f \colon A \rightarrow B$ and I want to get bijection. Can I just resting codomain to $f(A)$? I know that every function i...