Quadratic field

A quadratic field $𝐾$ is a number field of degree 2, #m/def/num/alg i.e. $[𝐾 : ℚ] = 2$ whence $𝐾 = ℚ (\sqrt 𝑑)$ for some squarefree $𝑑 \in ℤ$ .

Proof

Let ${1, 𝜗}$ be a $ℚ$ -basis for $𝐾$ , where without loss of generality $𝜗 \in O 𝐾$ is an algebraic integer, whence $𝜗 2 = 𝑎 𝜗 + 𝑏$ for some $𝑎, 𝑏 \in ℤ$ . Let $𝜑 = 2 𝜗 - 𝑎$ , so $𝜑 2 = 4 𝜗 2 - 4 𝑎 𝜗 + 𝑎 2 = 𝑎 2 + 4 𝑏$ , and clearly ${1, 𝜑}$ is also a $ℚ$ -basis for $𝐾$ . Setting $𝑎 2 + 4 𝑏 = 𝑘 2 𝑑$ where $𝑘, 𝑑 \in ℤ$ and $𝑑$ is squarefree, we have $\sqrt 𝑑 = 𝜑 / 𝑘$ , so $𝐾 = ℚ (\sqrt 𝑑)$ .

The ring of integers of a quadratic field are the Quadratic integers, whose structure is largely determined by $𝑑$ mod $4$ . Any number which is an element of a quadratic field is a quadratic number.

Properties

By quadratic integers, $𝐾$ is a monogenic field unless $𝑑 \equiv 4 1$ .

Classification

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