Editing Field (mathematics) (section)

=== Subfields and prime fields<span class="anchor" id="Prime field"></span> ===
A ''[[field extension|subfield]]'' {{math|''E''}} of a field {{math|''F''}} is a subset of {{math|''F''}} that is a field with respect to the field operations of {{math|''F''}}. Equivalently {{math|''E''}} is a subset of {{math|''F''}} that contains {{math|1}}, and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. This means that {{math|1 ∊ ''E''}}, that for all {{math|''a'', ''b'' ∊ ''E''}} both {{math|''a'' + ''b''}} and {{math|''a'' ⋅ ''b''}} are in {{math|''E''}}, and that for all {{math|''a'' ≠ 0}} in {{math|''E''}}, both {{math|−''a''}} and {{math|1/''a''}} are in {{math|''E''}}.

[[Field homomorphism]]s are maps {{math|''φ'': ''E'' → ''F''}} between two fields such that {{math|1=''φ''(''e''<sub>1</sub> + ''e''<sub>2</sub>) = ''φ''(''e''<sub>1</sub>) + ''φ''(''e''<sub>2</sub>)}}, {{math|1=''φ''(''e''<sub>1</sub>''e''<sub>2</sub>) = ''φ''(''e''<sub>1</sub>)&hairsp;''φ''(''e''<sub>2</sub>)}}, and {{math|1=''φ''(1<sub>''E''</sub>) = 1<sub>''F''</sub>}}, where {{math|''e''<sub>1</sub>}} and {{math|''e''<sub>2</sub>}} are arbitrary elements of {{math|''E''}}. All field homomorphisms are [[injective]].<ref>{{harvp|Adamson|2007|loc=§I.3}}</ref> If {{math|''φ''}} is also [[surjective]], it is called an [[isomorphism]] (or the fields {{math|''E''}} and {{math|''F''}} are called isomorphic).

A field is called a '''prime field''' if it has no proper (i.e., strictly smaller) subfields. Any field {{math|''F''}} contains a prime field. If the [[Characteristic (algebra)|characteristic]] of {{math|''F''}} is {{math|''p''}} (a prime number), the prime field is isomorphic to the finite field {{math|'''F'''<sub>''p''</sub>}} introduced below. Otherwise the prime field is isomorphic to {{math|'''Q'''}}.<ref>{{harvp|Adamson|2007|loc=p. 12}}</ref>