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  "body": "\n## Summary\n\nLet $L$ be a finite extension of a number field $K$.\nA prime ideal $\\mathfrak{p}$ of $K$ ramifies in $L$ when its factorisation in $L$ has a repeated prime factor.\nThe exponent of a prime factor is its ramification index.\nOnly finitely many primes ramify, namely those dividing the relative discriminant.\nOver $\\Q$ these are the primes dividing the [[discriminant|E0VZTNPJXA]] of $L$.\nRamification is one part of the [[splitting of prime ideals|B7KNZCBFYT]], and the word comes from the branching of covering maps of Riemann surfaces.\n\n## Explanations\n\n### L1 human\n\nWhen a prime number is factored in a larger ring of integers, the factors may come with exponents bigger than $1$.\nThis is ramification: two or more factors have merged into one.\nIn $\\Z[i]$, the prime $2$ factors as $-i(1+i)^2$, so $2$ ramifies with index $2$.\nEvery other prime number is unramified in $\\Z[i]$.\nA picture: the map $z \\mapsto z^2$ on the complex plane is two-to-one except at $0$, where the two preimages come together; this point is a branch point.\n",
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