Alias Email
How do I remove an alias email address from my Apple ID?
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It depends what you mean by 'alias': Apple use the term for two slightly different things.
An 'email alias' is an additional address (not account) within your iCloud account. It has different text before the '@' and is usually an @icloudcom address. You can remove one by going to the iCloud web page, clicking on the cogwheel icon at bottom left, choosing Accounts, highlighting the alias in the list and clicking the minus sign at the bottom of the list.
However: if you were a member of MobileMe before it closed in 2012 you have an @me.com address where the text before the '@' is the same as your principal @icloud.com address. Similarly if you go far enought back to have been a member of DotMac you have the @mac.com variant. These variants of the main address cannot be removed.