= Quadratic field
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= Quadratic number field
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A quadratic number field is a <number field> of <degree of a field extension> two, necessarily of the form $\mathbb Q(\sqrt d)$ for a <square-free integer> $d\ne0,1$. Its two <field embeddings> send $a+b\sqrt d$ to $a\pm b\sqrt d$. If $d>0$, both images lie in the <real numbers>, giving a <totally real number field>; if $d<0$, they form a pair related by <complex conjugation>, giving an <imaginary quadratic field>. The <ring of integers of a quadratic field> depends on $d$ modulo four.https://kconrad.math.uconn.edu/blurbs/gradnumthy/quadraticgrad.pdf[Keith Conrad, Factoring in Quadratic Fields, §§1–3]{ref}
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