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.

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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.