hola.... probaste con

return response('hola');

this is how I do it

Form

<form class="form-horizontal ajax-form" role="form" method="POST" action="{{ url('/api/v1/user') }}">
<input type="hidden" name="_token" value="{{ csrf_token() }}">
<input type="email" class="form-control" name="email" value="">
<input type="password" class="form-control" name="password">
<span class="showError"></span>
<button type="submit" class="btn btn-primary btnSubmit">Save</button>
</form>


#script
<script>

$(".ajax-form").submit(function(e) 
{
    e.preventDefault();

    $('.btnSubmit').attr('disabled','disabled');//disabled submit button

	var form = $(this);
    var url = form.attr('action');
    var method = form.find('input[name="_method"]').val() || 'POST' ;
    var data= form.serialize();
    
    $.ajax({
        url : url,
        type: method,
        data: data,
        dataType: 'json',
        success: function( json )
        {       
            $('.ajax-form')[0].reset(); //clear form
            $('.btnSubmit').removeAttr('disabled'); //enable button
        },
    	error   : function ( jqXhr, json, errorThrown ) 
		{
			var errors = jqXhr.responseJSON;//get Larevel errors

	        if( jqXhr.status == 422)  //  validation failed
	        {    
            	var errorsHtml ='';
             	$.each( errors, function( key, value ) {
                	errorsHtml += '<li>' + value[0] + '</li>'; 
            	});
            	$('.showError').html( errorsHtml );
	        }
	        $('.btnSubmit').removeAttr('disabled');
        }
    });                
});
</script>


# Controller
public function store(Request $request)
{
   $v = Validator::make( $request->all() , [
        'email'     => 'required|email|max:255|unique:users',
        'password'  => 'required|min:6',            
        ]);
    
    if( $v->fails() )
    {   
         return response( $v->errors() ,422); 
    }

    $user = new User;
    $user->email = $request->email;
    $user->password = bcrypt( $request->password );
    $user->save();

    return response( [ 'success'=>true, 'data' => $user ] , 200);
}

Still does not work and do not understand. Everywhere says it works well.

did you insert jquery?

<script src="//cdnjs.cloudflare.com/ajax/libs/jquery/2.1.3/jquery.min.js"></script>

I changed your form a bit and it works

<script src="//cdnjs.cloudflare.com/ajax/libs/jquery/2.1.4/jquery.min.js"></script>
<form class="ajax-form" method="POST" action="{{ url('/login') }}">
<input type="hidden" name="_token" id="token" value="{{ csrf_token() }}">
<div class="form-group">
<input type="text" placeholder="Usuario" name="user" id="email" class="form-control">
</div>
<div class="form-group">
<input type="password" placeholder="Contraseña" name="password" id="pass" class="form-control">
</div>
<button type="submit" class="" id="login">Save</button>
</form>

<script>

$(".ajax-form").submit(function(e) 
{
    e.preventDefault();
    var form = $(this);
    var url = form.attr('action');
    var datos = form.serialize();   
    var logica  = Logica( url, datos );
});
 

function Logica(url, datos )
{

  $.ajax({
        type    : "POST",
        url     : url,
        data    : datos,
        dataType: 'json',
        success : function( json )
        {       
            alert( json );
        }

    });                

}

</script>

I included in my code jquery

<script type="text/javascript" src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.3/jquery.min....;></script>

After installing google chrome, gave me an error that was understood, the script is good, the error it gives is the following

XMLHttpRequest can not load http: //vivianaglonski.web/login. No 'Access-Control-Allow-Origin "header is present on the requested resource. Origin 'http: //vivianaglonski.prog' THEREFORE is not allowed access.

you need to enable CORS! or change the form action URL

#from 
action="http://vivianaglonski.web/";

# login to 

action="{{ url('/login') }}"