Ring of integers

Depends on

• Number field
• Integral closure in an extension ring and integrally closed domains

Used by

• Minimal-polynomial criterion for algebraic integers Corollary
• Rings of integers are Dedekind domains Corollary
• Total ramification at a prime-power cyclotomic level Corollary
• Unit ranks by signature Corollary
• Order in a number field Definition
• Primes above and residue degree Definition
• Quadratic Gauss sum in a prime cyclotomic field Definition
• S-integers and S-units of a number field Definition
• Integers of Q Example
• Real quadratic units and Pell's equation Example
• S-units of Q Example
• Two independent units in a real cubic field Example
• Units of ℚ and the imaginary quadratic fields Example
• A number-field unit is exactly an algebraic integer of norm plus or minus one Lemma
• Choice-free prime factorisation for a monogenic number ring Lemma
• Finitely many roots of unity in a number field Lemma
• Integral basis and discriminant of a coprime-discriminant compositum Lemma
• Kernel of the unit logarithm is the roots of unity Lemma
• Prime-power cyclotomic ring, discriminant support and p factor Lemma
• Clearing denominators for an algebraic number Theorem
• Integers in a quadratic field Theorem
• Integral ideal factorisation in a number field, in ZF Theorem
• Kronecker root-of-unity criterion Theorem
• Number field places classification Theorem
• Number-field integer rings and ideals are full lattices Theorem
• Product formula for a number field Theorem
• Ramification is detected by the number-field discriminant Theorem
• Ring of integers of every cyclotomic field Theorem
• The ring of integers has rank the degree Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Milne, Definition 2.5 (standard reference, not scraped)