# calculating the splitting of primes

Let $K|L$ be an extension^{} of number fields^{}, with rings of integers^{} ${\mathcal{O}}_{K},{\mathcal{O}}_{L}$. Since this extension is separable (http://planetmath.org/SeparablePolynomial), there exists $\alpha \in K$ with $L(\alpha )=K$ and by multiplying by a suitable integer, we may assume that $\alpha \in {\mathcal{O}}_{K}$ (we do not require that ${\mathcal{O}}_{L}[\alpha ]={\mathcal{O}}_{K}$. There is not, in general, an $\alpha \in {\mathcal{O}}_{L}$ with this property). Let $f\in {\mathcal{O}}_{L}[x]$ be the minimal polynomial^{} of $\alpha $.

Now, let $\U0001d52d$ be a prime ideal^{} of $L$ that does not divide $\mathrm{\Delta}(f)\mathrm{\Delta}{({\mathcal{O}}_{K})}^{-1}$, and let $\overline{f}\in {\mathcal{O}}_{L}/\U0001d52d{\mathcal{O}}_{L}[x]$ be the reduction^{} of $f$ mod $\U0001d52d$, and let $\overline{f}={\overline{f}}_{1}\mathrm{\cdots}{\overline{f}}_{n}$ be its factorization into irreducible polynomials^{}. If there are repeated factors, then $p$ splits in $K$ as the product

$$\U0001d52d=(\U0001d52d,{f}_{1}(\alpha ))\mathrm{\cdots}(\U0001d52d,{f}_{n}(\alpha )),$$ |

where ${f}_{i}$ is any polynomial^{} in ${\mathcal{O}}_{L}[x]$ reducing to ${\overline{f}}_{i}$. Note that in this case $\U0001d52d$ is unramified, since all ${f}_{i}$ are pairwise coprime mod $\U0001d52d$

For example, let $L=\mathbb{Q},K=\mathbb{Q}(\sqrt{d})$ where $d$ is a square-free integer.
Then $f={x}^{2}-d$. For any prime $\U0001d52d$, $f$ is irreducible^{} mod $\U0001d52d$ if and only if it has no roots mod $\U0001d52d$, i.e. $d$ is a quadratic non-residue mod $\U0001d52d$. Using quadratic reciprocity, we can obtain a congruence^{} condition mod $4p$ for which primes split and which do not. In general, this is possible for all fields with abelian^{} Galois groups^{}, using field .

Furthermore, let ${K}^{\prime}$ be the splitting field^{} of $L$. Then $G=\mathrm{Gal}({K}^{\prime}|L)$ acts on the roots of $f$, giving a map $G\to {S}_{m}$, where $m=\mathrm{deg}f$. Given a prime $\U0001d52d$ of ${\mathcal{O}}_{L}$, the Artin symbol^{} $[\U0001d513,{K}^{\prime}|L]$ for any $\U0001d513$ lying over $\U0001d52d$ is determined up to conjugacy by $\U0001d52d$. Its in ${S}_{n}$ is a product of disjoint cycles of length ${m}_{1},\mathrm{\dots},{m}_{n}$ where ${m}_{i}=\mathrm{deg}{f}_{i}$.
This is useful not just for prime splitting, but also for the calculation of Galois groups.

Another useful fact is the Frobenius theorem, which that every element of $G$ is $[\U0001d513,{K}^{\prime}|L]$ for infinitely many primes $\U0001d513$ of ${\mathcal{O}}_{{K}^{\prime}}$.

For example, let $f={x}^{3}+{x}^{2}+2\in \mathbb{Z}[x]$. This is irreducible mod 3, and thus irreducible. Galois theory^{} tells us that $G=\mathrm{Gal}({K}^{\prime}|L)$ is a subgroup^{} of ${S}_{3}$, and so is isomorphic^{} to ${C}_{3}$ or ${S}_{3}$, but it is not obvious which. But if we consider $p=7$, $f\equiv (x-2)({x}^{2}+3x-1)\phantom{\rule{veryverythickmathspace}{0ex}}(mod7)$, and the quadratic factor is irreducible mod 7. Thus, $G\cong {S}_{3}$.

Or let $f={x}^{4}+a{x}^{2}+b$ for some integers $a,b$ and is irreducible. For a prime $p$, consider the factorization of $f$. Either it remains irreducible ($G$ contains a 4-cycle), splits as the product of irreducible quadratics ($G$ contains a cycle of the form $(12)(34)$) or $\overline{f}$ has a root. If $\beta $ is a root of $f$, then so is $-\beta $, and so assuming $p\ne 2$, there are at least two roots, and so a 3-cycle is impossible. Thus $G\cong {C}_{4}$ or ${D}_{4}$.

Title | calculating the splitting of primes |
---|---|

Canonical name | CalculatingTheSplittingOfPrimes |

Date of creation | 2013-03-22 13:53:24 |

Last modified on | 2013-03-22 13:53:24 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 12 |

Author | mathcam (2727) |

Entry type | Topic |

Classification | msc 11R04 |

Related topic | PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ |

Related topic | PrimeIdealDecompositionInCyclotomicExtensionsOfMathbbQ |

Related topic | NumberField |

Related topic | SplittingAndRamificationInNumberFieldsAndGaloisExtensions |