The M4RIE library for dense linear algebra over small fields with even characteristic. We describe algorithms and implementations for linear algebra with dense matrices over 𝔽 2 e for 2≀e≀10. Our main contributions are: (1) a specialisation of precomputation tables to 𝔽 2 e , called Newton-John tables in this work, to avoid scalar multiplications in Gaussian elimination and matrix multiplication, (2) an efficient implementation of Karatsuba-style multiplication for matrices over extension fields of 𝔽 2 and (3) a description of an open-source library – called M4RIE – providing the fastest known implementation of dense linear algebra over 𝔽 2 e with 2≀e≀10.