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  "body": "\n## Summary\n\nLet $L/K$ be a Galois extension of number fields, and let $\\mathfrak{P}$ be a prime of $L$ that is unramified over $K$.\nLet $q$ be the size of the residue field of the prime of $K$ below $\\mathfrak{P}$.\nThe Frobenius element of $\\mathfrak{P}$ is the unique element of the [[decomposition group|7X92G8CQ33]] that acts on the residue field as the [[Frobenius automorphism|0CGCPX8Z70]] $x \\mapsto x^q$.\nIts conjugacy class depends only on the prime of $K$ below $\\mathfrak{P}$.\nFrobenius elements encode how primes split, and they are the input of the Artin map in [[class field theory|BEKCEYHCHX]] and of [[Chebotarev's density theorem|X8VKW5Y1CZ]].\n\n## Explanations\n\n### L1 human\n\nOver a finite field with $q$ elements, raising to the power $q$ is a symmetry of every larger finite field.\nA Frobenius element lifts this symmetry from the residue field to the number field itself.\nIn $\\Q(i)/\\Q$, the Frobenius element at an odd prime $p$ is the identity if $p \\equiv 1 \\pmod 4$ and complex conjugation if $p \\equiv 3 \\pmod 4$.\nThis matches whether $p$ splits or stays prime in $\\Z[i]$.\nSo one group element records the splitting of a prime.\n",
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