Abstract: The inverse Galois problem is a major question in mathematics. For a given base field and a given finite group G, one would like to list all Galois extensions L/F such that the Galois group of L/F is G.

In this work we shall solve this problem for all fields F , and for group G of unipotent 4 × 4 matrices over F_2 . We also list all 16 U_4 (F_2 )- extensions of Q_2 . The importance of these results is that they answer the inverse Galois problem in some specific cases. This is joint work with Jan Minac and Nguyen Duy Tan.