[Merged by Bors] - feat(number_theory/legendre_symbol/jacobi_symbol): add new file with the definition and properties of the Jacobi symbol #16395
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…definition and properties of the Jacobi symbol
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The title should be something like feat(number_theory/legendre_symbol/jacobi_symbol): part 1 |
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I had edited the title already, but for some reason the change seems to have been lost... |
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Since the notation introduced in the file leads to problems with list notation (see the comments near the beginning of |
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Sep 8, 2022
…the definition and properties of the Jacobi symbol (#16395) This PR consists of a new file, `number_theory.legendre_symbol.jacobi_symbol`, that contains a definition of the Jacobi symbol (and sets up notation `[a | b]ⱼ` for it) and then proves a number of results, e.g., multiplicativity in both arguments, expressions for the cases `a = -1`, `a = 2`, `a = -2`, and several versions of quadratic reciprocity. It also proves that the symbol depends on a only through its residue class `mod b` and that it depends on `b` only through its residue class `mod 4*a` (quadratic reciprocity is needed for the latter). The code has already been quite thoroughly reviewed while it was part of #16290 (which then was reduced to the auxiliary results needed for the Jacobi symbol code). Discussion on [Zulip](https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/Jacobi.20symbol/near/295816984)
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Sep 13, 2022
…the definition and properties of the Jacobi symbol (#16395) This PR consists of a new file, `number_theory.legendre_symbol.jacobi_symbol`, that contains a definition of the Jacobi symbol (and sets up notation `[a | b]ⱼ` for it) and then proves a number of results, e.g., multiplicativity in both arguments, expressions for the cases `a = -1`, `a = 2`, `a = -2`, and several versions of quadratic reciprocity. It also proves that the symbol depends on a only through its residue class `mod b` and that it depends on `b` only through its residue class `mod 4*a` (quadratic reciprocity is needed for the latter). The code has already been quite thoroughly reviewed while it was part of #16290 (which then was reduced to the auxiliary results needed for the Jacobi symbol code). Discussion on [Zulip](https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/Jacobi.20symbol/near/295816984)
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Sep 14, 2022
…ym* to other namespaces (#16461) This PR moves `legendre_sym` and `jacobi_sym` (and `qr_sign`) to the root namespace. It also moves definitions and lemmas named `legendre_sym_...` to the `legendre_sym` namespace and analogous for `jacobi_sym_...` and `qr_sign_...`. We change names like `jacobi_sym_two` to `jacobi_sym.at_two`. This mostly affects the files `number_theory.legendre_symbol.quadratic_reciprocity` and `number_theory.legendre_symbol.jacobi_symbol`. Note that names like `exists_sq_eq_neg_one_iff` have been left in `zmod`. We prefix `quadratic_reciprocity*` by `legendre_sym` to disambiguate with the versions for the Jacobi symbol. (The links in `docs/100.yaml` and `docs/overview.yaml` are updated accordingly.) We also move the results in `number_theory.legendre_symbol.gauss_eisenstein_lemmas` from the `legendre_symbol` namespace to the `zmod` namespace. See this [Zulip thread](https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/Jacobi.20symbol/near/297792676) and the comments at [#16395](#16395) for discussion.
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Sep 21, 2022
…the definition and properties of the Jacobi symbol (#16395) This PR consists of a new file, `number_theory.legendre_symbol.jacobi_symbol`, that contains a definition of the Jacobi symbol (and sets up notation `[a | b]ⱼ` for it) and then proves a number of results, e.g., multiplicativity in both arguments, expressions for the cases `a = -1`, `a = 2`, `a = -2`, and several versions of quadratic reciprocity. It also proves that the symbol depends on a only through its residue class `mod b` and that it depends on `b` only through its residue class `mod 4*a` (quadratic reciprocity is needed for the latter). The code has already been quite thoroughly reviewed while it was part of #16290 (which then was reduced to the auxiliary results needed for the Jacobi symbol code). Discussion on [Zulip](https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/Jacobi.20symbol/near/295816984)
b-mehta
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Sep 21, 2022
…ym* to other namespaces (#16461) This PR moves `legendre_sym` and `jacobi_sym` (and `qr_sign`) to the root namespace. It also moves definitions and lemmas named `legendre_sym_...` to the `legendre_sym` namespace and analogous for `jacobi_sym_...` and `qr_sign_...`. We change names like `jacobi_sym_two` to `jacobi_sym.at_two`. This mostly affects the files `number_theory.legendre_symbol.quadratic_reciprocity` and `number_theory.legendre_symbol.jacobi_symbol`. Note that names like `exists_sq_eq_neg_one_iff` have been left in `zmod`. We prefix `quadratic_reciprocity*` by `legendre_sym` to disambiguate with the versions for the Jacobi symbol. (The links in `docs/100.yaml` and `docs/overview.yaml` are updated accordingly.) We also move the results in `number_theory.legendre_symbol.gauss_eisenstein_lemmas` from the `legendre_symbol` namespace to the `zmod` namespace. See this [Zulip thread](https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/Jacobi.20symbol/near/297792676) and the comments at [#16395](#16395) for discussion.
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This PR consists of a new file,
number_theory.legendre_symbol.jacobi_symbol, that contains a definition of the Jacobi symbol (and sets up notation[a | b]ⱼfor it) and then proves a number of results, e.g., multiplicativity in both arguments, expressions for the casesa = -1,a = 2,a = -2, and several versions of quadratic reciprocity. It also proves that the symbol depends on a only through its residue classmod band that it depends onbonly through its residue classmod 4*a(quadratic reciprocity is needed for the latter).The code has already been quite thoroughly reviewed while it was part of #16290 (which then was reduced to the auxiliary results needed for the Jacobi symbol code).
Discussion on Zulip