Decree for the Reasonable Restriction of Underage Sorcery
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The Decree for the Reasonable Restriction of Underage Sorcery is a fictional law in the Harry Potter books by J.K. Rowling. According to this law, no underaged wizard (defined in the books as being below 17 in age) is allowed to perform any voluntary magic outside of their school. In other words, characters in the book are allowed to perform magic while in Hogwarts School of Witchcraft and Wizardry, as they would obviously have to do in order to learn it; however, they are forbidden to use magic while on vacation from school, the idea being that they could be seen by Muggles or cause harm to themselves or others.
Harry Potter runs afoul of this law twice within the course of the series. The first time, in Harry Potter and the Chamber of Secrets, was not actually his fault; the house-elf Dobby had performed magic in his house, and the Ministry of Magic, unable to detect the conjurer of the spell and only the location, assumed that he, the only wizard in his house, must have done it (which was Dobby's plan, to try to keep Harry from returning to school). Harry only received a warning letter in that case. The second time, in Harry Potter and the Order of the Phoenix, was far more serious; he had conjured a Patronus in order to ward off two dementors that threatened the safety of both his cousin and him, and was tried by the Wizengamot over whether or not he would remain at Hogwarts or be expelled, but was acquitted when Albus Dumbledore, coming to his defence, was able to prove that Harry had indeed been in a life-and-death situation (for which there is an exception to the rule).
Harry had performed magic a few times prior to these situations out of school, when he was younger and without knowing he was doing so or meaning to; he once also accidentally blew up his Aunt Marge like a balloon by mistake. However, because these were involuntary, he was not charged; it is unclear whether or not accidental magic is always excused under the Decree.
The books contain plenty of reports of students performing magic on the Hogwarts Express (for example members of the DA attacking Malfoy at the end of 'Harry Potter and the Order of the Phoenix' or Professor Slughorn being impressed by Ginny Weasley's 'Bat Bogey Hex' at the start of 'Harry Potter and the Half Blood Prince'). It is possible that the Hogwarts Express is classified as part of the school to make this possible (or it is possible that the decree cannot be enforced - see below).
In the 'Harry Potter and the Chamber of Secrets' film, Hermione Granger performs the 'oculus reparo' charm in Diagon Alley. It is likely this event was not noticed by the Ministry because the Ministry only knows where the magic is performed, and Diagon Alley would have many qualified wizards and witches performing minor magic.
In 'Harry Potter and the Half Blood Prince', Dumbledore gives Harry some information that provides some background about how the decree is enforced by the Ministry. According to Dumbledore, the Ministry know only where magic has been carried out, but not by whom. So if an underage wizard or witch performed magic within their parents home, the ministry would be unable to differentiate this from the parent performing magic, and therefore would be unable to detect or prove the violation. Therefore they usually rely on parents to enforce the decree. Sadly this fact prevented the ministry discovering Tom Riddles murder of his father during this book.