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Field and Galois

Field and Galois· L01

Fields & field extensions

What a field is, simple extensions, and degree.

Note

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Field extension

If K⊆LK \subseteq L are fields, then L/KL/K is a field extension. The degree [L:K][L:K] is dim⁡KL\dim_K L.

P1

Degree of a quadratic extension

Show that [Q(2):Q]=2[\mathbb{Q}(\sqrt{2}):\mathbb{Q}] = 2.

Solution · P1
  1. The minimal polynomial of 2\sqrt{2} over Q\mathbb{Q} is x2−2x^2 - 2 (irreducible by Eisenstein or parity).
  2. Hence {1,2}\{1, \sqrt{2}\} is a basis of Q(2)\mathbb{Q}(\sqrt{2}) over Q\mathbb{Q}, so the degree is 22.