Editing Algebraic element (section)

{{Short description|Concept in abstract algebra}}
In [[mathematics]], if {{math|''A''}} is an [[associative algebra]] over {{math|''K''}}, then an element {{math|''a''}} of {{math|''A''}} is an '''algebraic element''' over {{math|''K''}}, or just '''algebraic over''' {{math|''K''}}, if there exists some non-zero [[polynomial]] <math>g(x) \in K[x]</math> with [[coefficient]]s in {{math|''K''}} such that {{math|''g''(''a'') {{=}} 0}}.<ref>{{Cite book |last=Roman |first=Steven |title=Advanced Linear Algebra |date=2008 |publisher=Springer New York Springer e-books |isbn=978-0-387-72831-5 |series=Graduate Texts in Mathematics |location=New York, NY |pages=458–459 |chapter=18}}</ref> Elements of {{math|''A''}} that are not algebraic over {{math|''K''}} are '''transcendental over''' {{math|''K''}}. A special case of an associative algebra over <math>K</math> is an [[extension field]] <math>L</math> of <math>K</math>.

These notions generalize the [[algebraic number]]s and the [[transcendental number]]s (where the field extension is {{math|'''C'''/'''Q'''}}, with {{math|'''C'''}} being the field of [[complex number]]s and {{math|'''Q'''}} being the field of [[rational number]]s).