I have to determine the equations of motion for both the complex scalar field $\varphi$ and the electromagnetic field $A_\mu$ by using the Euler-Lagrange equations.

Now I know, that because the scalar field is complex it has twice the degrees of freedom so I get two equations of motion (?). They should be given by: $$\frac{\partial \mathcal{L}}{\partial\varphi}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu \varphi)}=0$$

Yes.

and $$\frac{\partial \mathcal{L}}{\partial\varphi^*}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu \varphi^*)}=0.$$

Yes.

For the electromagnetic field $A_\mu$ it should just be: $$\frac{\partial \mathcal{L}}{\partial A_\mu}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\mu)}=0.$$

No, this one above is wrong. (And you can see this by the fact that the indexing doesn't make sense... there is a summation convention, but three of the indices on the second term have the same symbol...)

You get a separate equation for each of the four $A_\nu$: $$\frac{\partial \mathcal{L}}{\partial A_\nu}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\nu)}=0.$$

I have to determine the equations of motion for both the complex scalar field $\varphi$ and the electromagnetic field $A_\mu$ by using the Euler-Lagrange equations.

Now I know, that because the scalar field is complex it has twice the degrees of freedom so I get two equations of motion (?). They should be given by: $$\frac{\partial \mathcal{L}}{\partial\varphi}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu \varphi)}=0$$

Yes.

and $$\frac{\partial \mathcal{L}}{\partial\varphi^*}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu \varphi^*)}=0.$$

Yes.

For the electromagnetic field $A_\mu$ it should just be: $$\frac{\partial \mathcal{L}}{\partial A_\mu}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\mu)}=0.$$

No, this one quoted above is wrong. (And you can see this by the fact that the indexing doesn't make sense... there is a summation convention, but three of the indices on the second term have the same symbol...)

You get a separate equation for each of the four $A_\nu$: $$\frac{\partial \mathcal{L}}{\partial A_\nu}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\nu)}=0.$$

I have to determine the equations of motion for both the complex scalar field $\varphi$ and the electromagnetic field $A_\mu$ by using the Euler-Lagrange equations.

Now I know, that because the scalar field is complex it has twice the degrees of freedom so I get two equations of motion (?). They should be given by: $$\frac{\partial \mathcal{L}}{\partial\varphi}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu \varphi)}=0$$

Yes.

and $$\frac{\partial \mathcal{L}}{\partial\varphi^*}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu \varphi^*)}=0.$$

Yes.

For the electromagnetic field $A_\mu$ it should just be: $$\frac{\partial \mathcal{L}}{\partial A_\mu}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\mu)}=0.$$

No, this one above is wrong. (And you can see this by the fact that the indexing doesn't make sense... there is a summation convention, but three of the indices on the second term have the same symbol...)

You get a separate equation for each of the four $A_\nu$: $$\frac{\partial \mathcal{L}}{\partial A_\nu}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\nu)}=0.$$

I have to determine the equations of motion for both the complex scalar field $\varphi$ and the electromagnetic field $A_\mu$ by using the Euler-Lagrange equations.

Now I know, that because the scalar field is complex it has twice the degrees of freedom so I get two equations of motion (?). They should be given by: $$\frac{\partial \mathcal{L}}{\partial\varphi}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu \varphi)}=0$$

Yes.

and $$\frac{\partial \mathcal{L}}{\partial\varphi^*}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu \varphi^*)}=0.$$

Yes.

For the electromagnetic field $A_\mu$ it should just be: $$\frac{\partial \mathcal{L}}{\partial A_\mu}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\mu)}=0.$$

No, this one above is wrong. (And you can see this by the fact that the indexing doesn't make sense... there is a summation convention, but three of the indices on the second term have the same symbol...)

You get a separate equation for each of the four $A_\nu$: $$\frac{\partial \mathcal{L}}{\partial A_\nu}-\partial_\mu\frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\nu)}=0.$$