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5 November 2023
- 19:5319:53, 5 November 2023 diff hist +277 N Definition talk:Ring of Integers of Number Field Created page with "Indeed, the Ring of Integers is usually defined for algebraic number fields. This is what Neukrich uses. You can do it for all number fields, but nobody cares about the ring of integers of Q(π)... ~~~~"
- 13:3613:36, 5 November 2023 diff hist +109 m Definition:Free Abelian Group No edit summary
- 13:1013:10, 5 November 2023 diff hist +1 m Definition:Free Abelian Group No edit summary
- 13:1013:10, 5 November 2023 diff hist −108 Definition:Free Abelian Group No edit summary
- 13:0613:06, 5 November 2023 diff hist +76 Definition:Free Abelian Group No edit summary
- 12:5612:56, 5 November 2023 diff hist +23 Definition:Complex Embedding No edit summary current
- 12:5512:55, 5 November 2023 diff hist +30 Definition:Real Embedding No edit summary current
- 12:3812:38, 5 November 2023 diff hist +15 m Definition:Integral Basis No edit summary current
- 07:5207:52, 5 November 2023 diff hist +58 m Definition:Free Module No edit summary
- 07:5107:51, 5 November 2023 diff hist +2 m Definition:Free Abelian Group No edit summary
- 07:5107:51, 5 November 2023 diff hist +1 m Definition:Free Abelian Group No edit summary
- 07:5007:50, 5 November 2023 diff hist +38 m Definition:Free Abelian Group No edit summary
- 07:4907:49, 5 November 2023 diff hist +431 N Definition:Free Abelian Group Created page with "== Definition == Let $G$ be an abelian group. $G$ is a '''free abelian group''' if and only if it has an integral basis. That is, $G$ is a '''free abelian group''' if and only if the $\mathbb Z$-module associated with $G$ is a free $\mathbb Z$-module."
- 07:4507:45, 5 November 2023 diff hist +1 m Definition:Integral Basis No edit summary
- 07:4107:41, 5 November 2023 diff hist +298 N Definition:Integral Basis Created page with "== Definition == Let $(G,+)$ be an abelian group. Let $(G,+,\circ)$ be the $\mathbb Z$-module associated with $G$. An '''integral basis''' of $G$ is a basis of $G$ over $\mathbb Z$."
- 06:0206:02, 5 November 2023 diff hist −138 m Ring of Integers of Number Field is Free Z-Module No edit summary
- 06:0106:01, 5 November 2023 diff hist +35 m Ring of Integers of Number Field is Free Z-Module No edit summary
- 05:5405:54, 5 November 2023 diff hist −11 m Ring of Integers of Number Field is Free Z-Module No edit summary
- 05:5405:54, 5 November 2023 diff hist +496 N Ring of Integers of Number Field is Free Z-Module Created page with "== Theorem == Let $K$ be an algebraic number field. Let $\OO_K$ be its ring of integers. Then $\OO_K$ is a free $\mathbb Z$-module. === Dimension === The dimension of $\OO_K$ as a free $\mathbb Z$-module is equal to $[K:\mathbb Q]$. == Proof == {{proof wanted}} Category:Alge..."
- 05:4205:42, 5 November 2023 diff hist +56 Definition:Ring of Integers of Number Field No edit summary
- 05:2905:29, 5 November 2023 diff hist −10 m Definition:Discriminant No edit summary Tag: Manual revert
- 05:2905:29, 5 November 2023 diff hist −10 m Definition:Discriminant nvm
- 05:2505:25, 5 November 2023 diff hist +11 m Ring of Integers of Number Field is Dedekind Domain No edit summary
- 05:2405:24, 5 November 2023 diff hist +21 m Definition:Ring of Integers of Number Field No edit summary
- 05:1705:17, 5 November 2023 diff hist +20 m Definition:Discriminant No edit summary
- 05:1405:14, 5 November 2023 diff hist +435 N Definition:Complex Embedding Created page with "==Definition== Let $K$ be a subfield of $\mathbb C$. A '''complex embedding''' of $K$ is an embedding of $K$ into $\mathbb C$. That is, a '''complex embedding''' is an (injective) field homomorphism $\sigma:K\to\mathbb C$. ==Sources== {{Planetmath|title = real and complex embeddings|url = realandcomplexembeddings}}"
- 05:0805:08, 5 November 2023 diff hist 0 m Definition:Real Embedding No edit summary
- 05:0705:07, 5 November 2023 diff hist +10 m Definition:Real Embedding No edit summary
- 05:0705:07, 5 November 2023 diff hist +419 N Definition:Real Embedding Created page with "==Definition== Let $K$ be a subfield of $\mathbb C$. A '''real embedding''' of $K$ is an embedding of $K$ into $\mathbb R$. That is, a '''real embedding''' is an (Definition:Injective) field homomorphism $\sigma:K\to\mathbb R$. ==Sources== {{Planetmath|title = real and complex embeddings|url = realandcomplexembeddings}}"
- 04:2604:26, 5 November 2023 diff hist +542 N Definition talk:Field Homomorphism Created page with "I propose to change the definition Field Homomorphism to preserve multiplicative identity. Fields are unital rings, so it seems strange to not force a field homomorphism to be a unital ring homomorphism. Under this definition, there is an annoying edge case of the trivial homomorphism (all others are injective), which doesn't behave well and nobody cares about anyway... If nobody responds within a few days I will change the definition and related theorems. ~~~~"