We are interested in the analogue of a result proved in the number field case by E. Brown and C.J. Parry and in the function field case in odd characteristic by Zhang Xianke. Precisely, we study the ideal class number one problem for imaginary quartic Galois extensions of $k=\fq(x)$ of Galois group $\integers /2\integers \times\integers /2\integers$ in even characteristic.\\Let $L/k$ be such an extension and let $K_1$, $K_2$ and $K_3$ be the distinct subfields extensions of $L/k$. In even characteristic, the fields $K_i$ are Artin-Schreier extensions of $k$ and $L$ is the compositum of any two of them.\\Using the factorization of the zeta functions of this fields, we get a formula between their ideal class numbers which enables us to find all imaginary quartic Galois extensions $L/k$ of Galois group $\integers /2\integers \times\integers /2\integers$ with ideal class number one.