When you refer to the alarm icon, I assume you mean the one that appears at the top of the screen along with signal strength, battery etc to indicate that an alarm is set? AFAIK, a calendar reminder doesn't show there and simply appears as a notification on the pull-down when it goes off, with a sound if that's enabled.
In the first instance, you could try setting another alarm and see what appears in the staus bar - is it a second clock alarm icon, or does it use the same one? Is it a different icon? (if it is, the alarm isn't coming from the clock). If it's the same, and doesn't add an icon, now cancel the alarm you just set. If we're very lucky, that'll fix it. If not, then clearing data in the clock app would be the next step.
