Abstract Algebra: Theory and Applications · Problem 21.5.4

Finite Fields — Thomas Judson

Mathematics — Abstract Algebra · field-extension-basis-subfields

Problem

Consider the field extension Q(34,i)\mathbb{Q}(\sqrt[4]{3}, i) over Q\mathbb{Q}. (a) Find a basis for the field extension Q(34,i)\mathbb{Q}(\sqrt[4]{3}, i) over Q\mathbb{Q}. Conclude that [Q(34,i):Q]=8[\mathbb{Q}(\sqrt[4]{3}, i) : \mathbb{Q}] = 8. (b) Find all subfields FF of Q(34,i)\mathbb{Q}(\sqrt[4]{3}, i) such that [F:Q]=2[F : \mathbb{Q}] = 2. (c) Find all subfields FF of Q(34,i)\mathbb{Q}(\sqrt[4]{3}, i) such that [F:Q]=4[F : \mathbb{Q}] = 4.

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