feat(group_theory/torsion): Q/Z and Fq[X] are torsion groups #12310
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Julian
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eric-wieser
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eric-wieser
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Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
eric-wieser
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| /-- The group ℚ ⧸ ℤ. -/ | ||
| @[reducible] def qmodz := ℚ ⧸ (algebra.linear_map ℤ ℚ).range | ||
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| namespace qmodz | ||
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| open rat | ||
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| /-- ℚ ⧸ ℤ is a torsion group -/ | ||
| lemma is_torsion : add_monoid.is_torsion qmodz := | ||
| λ q, begin | ||
| induction q using quotient.induction_on', | ||
| simp_rw [is_of_fin_add_order_iff_nsmul_eq_zero, submodule.quotient.mk'_eq_mk, | ||
| ←submodule.quotient.mk_smul, submodule.quotient.mk_eq_zero, linear_map.mem_range, | ||
| algebra.linear_map_apply, algebra_map_int_eq, ring_hom.eq_int_cast, nsmul_eq_mul, mul_comm], | ||
| exact ⟨q.denom, q.pos, q.num, mul_denom_eq_num.symm⟩, | ||
| end |
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I think group_theory/specific_groups is probably a good place for this.
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| namespace polynomial | ||
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| open_locale polynomial | ||
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| variables {Fq : Type*} [field Fq] [fintype Fq] | ||
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| /-- Fq[X] is a torsion group -/ | ||
| lemma is_torsion : add_monoid.is_torsion Fq[X] := | ||
| λ f, begin | ||
| rw is_of_fin_add_order_iff_nsmul_eq_zero, | ||
| refine ⟨fintype.card Fq, fintype.card_pos, _⟩, | ||
| rw [nsmul_eq_smul_cast Fq _ f, finite_field.cast_card_eq_zero, zero_smul], | ||
| end | ||
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| end polynomial |
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This maybe belongs in field_theory.finite somewhere? Perhaps best to ask on Zulip.
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Is it reasonable to add these specific proofs of the existence of non-finite torsion groups? If so, is this the right spot to put that, or should they be distributed into
group_theory/specific_groups? mathlib doesn't seem to know much aboutQ/Zat the minute in particular.