Find the smallest positive integer value of $N$ such that field $K=\mathbb{Q}(\sqrt{N},\ \sqrt{i+1})$, where $i=\sqrt{-1}$, is Galois extension on $\mathbb{Q}$, then find the Galois group $Gal(K/\mathbb{Q}).$
1. **Determine the minimal polynomial of $\sqrt{1+i}$ over $\mathbb{Q}$:**

   Let $\alpha = \sqrt{1+i}$. Then $\alpha^2 = 1+i$. To find the minimal polynomial of $\alpha$, we need to eliminate $\alpha$ from the equation $\alpha^2 = 1+i$.

   \[
   \alpha^4 = (1+i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i
   \]

   Therefore, $\alpha^4 = 2i$. To find a polynomial with rational coefficients, we consider $\alpha^8$:

   \[
   \alpha^8 = (2i)^2 = -4
   \]

   Hence, the minimal polynomial of $\alpha$ over $\mathbb{Q}$ is:

   \[
   x^4 - 2x^2 + 2
   \]

2. **Identify the roots of the polynomial:**

   The roots of the polynomial $x^4 - 2x^2 + 2$ are $\pm \alpha$ and $\pm \beta$, where $\alpha = 2^{1/4} \zeta_{16}$ and $\beta = 2^{1/4} \zeta_{16}^{-1}$, with $\zeta_{16} = e^{2 \pi i / 16}$.

3. **Determine the splitting field:**

   The splitting field of the polynomial $x^4 - 2x^2 + 2$ is $\mathbb{Q}(\alpha, \beta)$. Note that $\alpha \beta = \sqrt{2}$, so $\sqrt{2} \in \mathbb{Q}(\alpha, \beta)$. This shows that $\mathbb{Q}(\alpha, \beta) = \mathbb{Q}(\sqrt{2}, \sqrt{1+i})$ is Galois over $\mathbb{Q}$.

4. **Show that $\alpha \not\in \mathbb{Q}(\beta)$:**

   Suppose, on the contrary, that $\alpha = a_0 + a_1 \beta + a_2 \beta^2 + a_3 \beta^3$ for some rationals $a_0, a_1, a_2, a_3$. Taking the complex conjugate, we have $\beta = a_0 + a_1 \alpha + a_2 \alpha^2 + a_3 \alpha^3$. Subtracting the two equations, we get:

   \[
   \alpha - \beta = a_1 (\beta - \alpha) + a_2 (\beta^2 - \alpha^2) + a_3 (\beta^3 - \alpha^3)
   \]

   Simplifying, we have:

   \[
   -1 - a_1 = a_2 (\beta + \alpha) + a_3 (\beta^2 + \alpha \beta + \alpha^2)
   \]

   Since $\beta + \alpha = 2^{1/4} \sqrt{2 + \sqrt{2}}$ and $\beta^2 + \alpha \beta + \alpha^2 = 2 + \sqrt{2}$, we have:

   \[
   -1 - a_1 = a_2 \cdot 2^{1/4} \sqrt{2 + \sqrt{2}} + a_3 (2 + \sqrt{2})
   \]

   The set $\{1, 2^{1/4}\sqrt{2 + \sqrt{2}}, 2 + \sqrt{2}\}$ is linearly independent over $\mathbb{Q}$, implying that $a_2 = 0$, $a_3 = 0$, and $a_1 = -1$. Therefore, $\alpha = a_0 - \beta$, or $\alpha + \beta = a_0 \in \mathbb{Q}$, which is a contradiction.

5. **Determine the smallest positive integer $N$:**

   From the above, the smallest such $N$ is $N = 2$.

6. **Find the Galois group $Gal(K/\mathbb{Q})$:**

   Let $L = \mathbb{Q}(\alpha, \beta) = \mathbb{Q}(\sqrt{2}, \alpha)$. Note that $L/\mathbb{Q}$ is a degree $8$ extension. Therefore, each automorphism $\sigma \in Gal(L / \mathbb{Q})$ is completely determined by $\sigma(\sqrt{2}) = \pm \sqrt{2}$ and $\sigma(\alpha) = \pm \alpha, \pm \sqrt{2}/\alpha$, all of which have order $2$. So the Galois group is:

   \[
   \mathbb{Z}/2 \times \mathbb{Z}/2 \times \mathbb{Z}/2
   \]

The final answer is $\boxed{\mathbb{Z}/2 \times \mathbb{Z}/2 \times \mathbb{Z}/2}$