I am considering these two sentences:
- I have no idea.
- I have no ideas.
Are they both correct, and if so, are they synonymous?
They are both correct and they do not mean the same thing.
"I have no idea" means "I do not know the answer to your particular question." It is about a complete lack of knowledge. It is a counterpart to a less common phrase "I have an idea what the answer is", which is used before answering a question when you think you know what the answer might be, but you are not sure that you are correct.
"I have no ideas" means "I am not creative and I have no original thoughts." It can also have the related meaning, if you are asked to come up with something to do, that "I cannot think of anything to do."
To have no idea is a phrase or idiom–depending which reference you consult–whose meaning is easily found in most online dictionaries. It has in fact two different meanings.
have no idea
(level) B1 informal to not know something:
- "Where's Serge?" "I've no idea."
- We had no idea when the power might come back on.
- "What do you think Bonnie would like for her birthday?" "I have absolutely no idea."
have no idea, not have the faintest, first, etc. idea (informal)
used to emphasize that you do not know something
“What's she talking about?” “I have no idea.”
He hasn't the faintest idea how to manage people.
I had no idea she'd had such a difficult life.
I don't have any idea where he is.
- to not know or understand
I have no idea what you're talking about.
- —used to reply to someone as an emphatic "yes"
"Was it hard?" "You have no idea (how hard it was)!"
Longman Dictionary concurs with the above and completely ignores the version with personal pronoun "I", and adds this detail
spoken used when you are telling someone that something is extremely good, bad etc
- You have no idea how worried I was.
Collins Dictionary's definition implies that it is most commonly used with the second person singular
you have no idea/you've no idea (phrase)
You can say you have no idea to emphasize how good or bad something is. [informal, emphasis]
- We are both so happy, you have no idea.
- You have no idea how depressed it made me
While the meaning of “I have no idea”, in its uncountable sense, is similar to “I don't have a clue” and the Victorian-sounding “I haven't got the foggiest”; the second phrase “I have no ideas", with the stress on no, would suggest the speaker lacked the ability and imagination to create something original.