Number field

Quadratic field

A quadratic field 𝐾 is a number field of degree 2, alg i.e. [𝐾 :ℚ] =2 whence 𝐾 =ℚ(√𝑑) for some squarefree 𝑑 ∈ℤ.

Proof

Let {1,𝜗} be a ℚ-basis for 𝐾, where without loss of generality 𝜗 ∈O𝐾 is an algebraic integer, whence 𝜗2 =𝑎𝜗 +𝑏 for some 𝑎,𝑏 ∈ℤ. Let 𝜑 =2𝜗 −𝑎, so 𝜑2 =4𝜗2 −4𝑎𝜗 +𝑎2 =𝑎2 +4𝑏, and clearly {1,𝜑} is also a ℚ-basis for 𝐾. Setting 𝑎2 +4𝑏 =𝑘2𝑑 where 𝑘,𝑑 ∈ℤ and 𝑑 is squarefree, we have √𝑑 =𝜑/𝑘, so 𝐾 =ℚ(√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

1. By quadratic integers, 𝐾 is a monogenic field unless 𝑑 ≡41.

Classification

• Real quadratic field
• Imaginary quadratic field

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