Abstract Algebra: Theory and Applications · Problem 22.4.4

Galois Theory — Thomas Judson

Mathematics — Abstract Algebra · finite-fields-construction

Problem

Let α\alpha be a zero of x3+x2+1x^3 + x^2 + 1 over Z2\mathbb{Z}_2. Construct a finite field of order 8. Show that x3+x2+1x^3 + x^2 + 1 splits in Z2(α)\mathbb{Z}_2(\alpha). Hint. There are eight elements in Z2(α)\mathbb{Z}_2(\alpha). Exhibit two more zeros of x3+x2+1x^3 + x^2 + 1 other than α\alpha in these eight elements.

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