Algebra over a field
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In abstract algebra, an algebra over a field F, or F-algebra is a ring A containing an isomorphic copy of F in the centre. Another way of expressing this is to say that A is a vector space over F equipped with a further algebraic structure of multiplication compatible with the vector space structure.
Examples
- Any extension field E/F can be regarded as an F-algebra.
- The matrix ring Mn(F) of n×n square matrices with entries in F is an F-algebra, with F embedded as the scalar matrices.
- The quaternion ring is an algebra over the field of real numbers.
References
- Serge Lang (1986). Algebraic number theory. Springer-Verlag. ISBN 0-387-94225-4.