Basic Alarm Clock function

Projects General Guidance
1 
//www.elegoo.com
//2016.12.08

int buzzer = 12;//the pin of the active buzzer
int button = 11;


void setup()
{
 pinMode(buzzer,OUTPUT);//initialize the buzzer pin as an output
 pinMode(button,INPUT_PULLUP); // initialize the button
}
void loop()
{
 unsigned char i;
 while(digitalRead(button == HIGH))
 {
   //output an frequency
   for(i=0;i<10;i++)
   {
    digitalWrite(buzzer,HIGH);
    delay(1);//wait for 1ms
    digitalWrite(buzzer,LOW);
    delay(1);//wait for 1ms
    }
    //output another frequency
     for(i=0;i<10;i++)
      {
        digitalWrite(buzzer,HIGH);
        delay(2);//wait for 2ms
        digitalWrite(buzzer,LOW);
        delay(2);//wait for 2ms
      }
      
  }
}

I am trying to replicate an alarm clock snooze button. My basic programming logic would tell me this would work. I'm very new to arduino though. What might I change or look at differently in order to stop the buzzing once the button is pushed down.

2

Instead of

while(digitalRead(button == HIGH))

try

while(digitalRead(button) == HIGH)