  In this paper, the K_2 group of F_2 coefficients group algebra of a
noncommutative group with 8 elements(dihedral group D_4 ) is calculated,which
is divided into three parts:The first part is the introduction of basic
knowledge related to algebra K-theory, and a method of Magurn to calculate
finite field coefficients noncommutative finite group algebra in reference [2].
In the second part, operation laws of Dennis-Stein symbols is introduced, and
we combined it with the fact that F_2[D_4] is a local ring to determind the
direct sum term of K_2(F_2[D_4]) can only be Z_2 or Z_4. In the third part, we
continue to make use of the fact that F_2[D_4] is a local ring, and proved that
the group D_1(F_2[D_4]) is an abelian group closely related to the group
K_2(F_2[D_4]) through operating Dennis-Stein symbols. Then, we used group
homology and the Kunneth formula of the finite abelian group version to
calculate all cases of H_2(D_1(F_2[D_4]),Z) , and substituted the obtained
results into the long exact sequence derived from the Hochschild-Serre spectral
sequence for testing, and finally constructed the result: K_2(F_2[D_4])=Z_2..
