Nicki Minaj had a well-publicized beef with which fellow "American Idol" judge during her employment as such on the show in 2012?

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Answer: Mariah Carey
More information: The American Idol spotlight was apparently too small for both Minaj and Mariah Carey, so the two had a popular tiff from 2012 to 2013 which more or less ended when they both left the show after only one season.