"The Irish prime minister."
If you wanted to phrase it that way, the correct version would be Taoiseach na hÉireann.
That phrase uses the genitive case which hasn't been taught in the course yet - if you're new to Irish and haven't encountered the genitive yet then please ignore all that follows and wait till you get the Genitive Case skill!
(In a phrase like this, there can be only one definite article. If both nouns have definite articles and the second noun is in the genitive, the rule is that you drop the first definite article.
The owner of the company = úinéir an chomhlachta The man's hat = the hat of the man = hata an fhir)