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Let $\mathbb{K}$ be an imaginary quadratic field such that $2$ splits intotwo primes $\mathfrak{p}$ and $\bar{\mathfrak{p}}$. Let $\mathbb{K}_{\infty}$be the unique $\mathbb{Z}_2$-extension of $\mathbb{K}$ unramified outside$\mathfrak{p}$. Let $\mathfrak{f}$ be an ideal coprime to $\mathfrak{p}$ and$\mathbb{L}$ be an arbitrary extension of $\mathbb{K}$ contained in the rayclass field $\mathbb{K}(\mathfrak{p}^2\mathfrak{f})$. Let$\mathbb{L}_{\infty}=\mathbb{K}_{\infty}\mathbb{L}$ and let $\mathbb{M}$ be themaximal $p$-abelian, $\mathfrak{p}$-ramified extension of$\mathbb{L}_{\infty}$. We set $X=Gal(\mathbb{M}/\mathbb{L}_{\infty})$. In thispaper we prove the Iwasawa main conjecture for the module $X$.

The Main Conjecture for Imaginary quadratic fields for the split prime $p=2$