Editing Algebra over a field (section)

== Classification of low-dimensional unital associative algebras over the complex numbers ==

Two-dimensional, three-dimensional and four-dimensional unital associative algebras over the field of complex numbers were completely classified up to isomorphism by [[Eduard Study]].<ref>{{ citation | last=Study | first=E. | year=1890 | title=Über Systeme complexer Zahlen und ihre Anwendungen in der Theorie der Transformationsgruppen | journal=Monatshefte für Mathematik | volume=1 |issue=1 | pages=283–354 | doi=10.1007/BF01692479 | s2cid=121426669 }}</ref>

There exist two such two-dimensional algebras. Each algebra consists of linear combinations (with complex coefficients) of two basis elements, 1 (the identity element) and ''a''. According to the definition of an identity element,
:<math>\textstyle 1 \cdot 1 = 1 \, , \quad 1 \cdot a = a \, , \quad a \cdot 1 = a \, . </math>
It remains to specify
:<math>\textstyle a a = 1 </math> &nbsp; for the first algebra,
:<math>\textstyle a a = 0 </math> &nbsp; for the second algebra.

There exist five such three-dimensional algebras. Each algebra consists of linear combinations of three basis elements, 1 (the identity element), ''a'' and ''b''. Taking into account the definition of an identity element, it is sufficient to specify
:<math>\textstyle a a = a \, , \quad b b = b \, , \quad a b = b a = 0 </math> &nbsp; for the first algebra,
:<math>\textstyle a a = a \, , \quad b b = 0 \, , \quad a b = b a = 0 </math> &nbsp; for the second algebra,
:<math>\textstyle a a = b \, , \quad b b = 0 \, , \quad a b = b a = 0 </math> &nbsp; for the third algebra,
:<math>\textstyle a a = 1 \, , \quad b b = 0 \, , \quad a b = - b a = b </math> &nbsp; for the fourth algebra,
:<math>\textstyle a a = 0 \, , \quad b b = 0 \, , \quad a b = b a = 0 </math> &nbsp; for the fifth algebra.
The fourth of these algebras is non-commutative, and the others are commutative.