Generally, you would capitalize "American History" when:
* It's used as the name of a specific course: "I'm taking American History this semester."
* It's part of a title: "The Impact of the Civil War on American History"
* It's used as a proper noun referring to a specific field of study: "She is a professor of American History."
You would not capitalize "American History" when:
* It's used as a general term: "We learned about American history in school."
* It's part of a phrase that is not a proper noun: "The study of American history is important."
Style Guides:
* The Chicago Manual of Style: Generally follows the rules outlined above.
* The MLA Handbook: Generally follows the rules outlined above.
* The AP Stylebook: Generally avoids capitalization unless it's part of a proper noun.
Example:
* Incorrect: I'm taking american history this semester.
* Correct: I'm taking American History this semester.
* Incorrect: The book covered the history of american history.
* Correct: The book covered the history of American history.
Best practice:
If you're unsure, consult the specific style guide for your paper. If no specific guide is provided, follow the general rules outlined above.