Editing Algebraic number theory (section)

===Real and complex embeddings===
Some number fields, such as {{math|'''Q'''(√{{Overline|2}})}}, can be specified as subfields of the real numbers. Others, such as {{math|'''Q'''(√{{Overline|&minus;1}})}}, cannot. Abstractly, such a specification corresponds to a field homomorphism {{math|''K'' → '''R'''}} or {{math|''K'' → '''C'''}}. These are called '''real embeddings''' and '''complex embeddings''', respectively.

A real quadratic field {{math|'''Q'''(√{{Overline|''a''}})}}, with {{math|''a'' ∈ '''Q''', ''a'' > 0}}, and {{math|''a''}} not a [[square number|perfect square]], is so-called because it admits two real embeddings but no complex embeddings. These are the field homomorphisms which send {{math|√{{Overline|''a''}}}} to {{math|√{{Overline|''a''}}}} and to {{math|&minus;√{{Overline|''a''}}}}, respectively. Dually, an imaginary quadratic field {{math|'''Q'''(√{{Overline|&minus;''a''}})}} admits no real embeddings but admits a conjugate pair of complex embeddings. One of these embeddings sends {{math|√{{Overline|&minus;''a''}}}} to {{math|√{{Overline|&minus;''a''}}}}, while the other sends it to its [[complex conjugate]], {{math|&minus;√{{Overline|&minus;''a''}}}}.

Conventionally, the number of real embeddings of {{math|''K''}} is denoted {{math|''r''<sub>1</sub>}}, while the number of conjugate pairs of complex embeddings is denoted {{math|''r''<sub>2</sub>}}. The '''signature''' of ''K'' is the pair {{math|(''r''<sub>1</sub>, ''r''<sub>2</sub>)}}. It is a theorem that {{math|1=''r''<sub>1</sub> + 2''r''<sub>2</sub> = ''d''}}, where {{math|''d''}} is the degree of {{math|''K''}}.

Considering all embeddings at once determines a function <math>M \colon K \to \mathbf{R}^{r_1} \oplus \mathbf{C}^{r_2}</math>, or equivalently <math>M \colon K \to \mathbf{R}^{r_1} \oplus \mathbf{R}^{2r_2}.</math>
This is called the '''Minkowski embedding'''.

The subspace of the codomain fixed by complex conjugation is a real vector space of dimension {{math|''d''}} called [[Minkowski space (number field)|Minkowski space]]. Because the Minkowski embedding is defined by field homomorphisms, multiplication of elements of {{math|''K''}} by an element {{math|''x'' ∈ ''K''}} corresponds to multiplication by a [[diagonal matrix]] in the Minkowski embedding. The [[dot product]] on Minkowski space corresponds to the trace form <math>\langle x, y \rangle = \operatorname{Tr}(xy)</math>.

The image of {{math|''O''}} under the Minkowski embedding is a {{math|''d''}}-dimensional [[lattice (group)|lattice]]. If {{math|''B''}} is a basis for this lattice, then {{math|det ''B''<sup>T</sup>''B''}} is the '''discriminant''' of {{math|''O''}}. The discriminant is denoted {{math|&Delta;}} or {{math|''D''}}. The covolume of the image of {{math|''O''}} is <math>\sqrt{|\Delta|}</math>.