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 series 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 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 holiday from school, to prevent it being seen by Muggles, and to prevent the children harming themselves or others accidentally.
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 a case of mistaken identity; 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 himself, 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 inflated his Aunt Marge 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, or if they can necessarily detect it.
The books contain plenty of reports of students performing magic on the Hogwarts Express (for example members of Dumbledore's Army attacking Draco Malfoy at the end of 'Harry Potter and the Order of the Phoenix' or Horace 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 '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. The inability to properly detect the culprit was one reason for the Ministry gaoling the wrong man for the murder of the Riddle family. It is not clear how the ministry is able to enforce the law effectively because of this; many characters perfomr magic in areas welll away from their own homes and in muggle areas without getting caught. It can also be presumed that some advanced magic cannot be detected by the ministry: Voldemort was able to conceal his Horcruxes without being detected.