feat(field_theory/finite): cardinality of images of polynomials #1554
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jcommelin
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Oct 16, 2019
| ... = _ : by rw [sum_const, add_monoid.smul_eq_mul', nat.cast_id] | ||
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| /-- If `f` and `g` are quadratic polynomials, then the `f.eval a + g.eval b = 0` has a solution. -/ | ||
| lemma exists_root_sum_quadratic {f g : polynomial α} (hf2 : degree f = 2) |
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This seems closely related to the very useful https://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem but I don't think it's a consequence. I certainly think that the lemma you formalised is useful outside of your direct application.
Would it make sense to have a version that uses the characteristic of alpha instead of its cardinality mod 2?
jcommelin
reviewed
Oct 16, 2019
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| open finset polynomial | ||
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| /-- The cardinality of a field is at most n times the cardinality of the image of a degree n |
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Suggested change
| /-- The cardinality of a field is at most n times the cardinality of the image of a degree n | |
| /-- The cardinality of a finite integral domain is at most n times the cardinality of the image of a degree n |
src/field_theory/finite.lean
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| rcases hd with ⟨x, ⟨a, _, ha⟩, ⟨b, _, hb⟩⟩, | ||
| use [a, b], | ||
| rw [ha, ← hb, eval_neg, neg_add_self] | ||
| end |
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This proof is long. But if you think you can't really split of useful sublemmas then I'm fine with merging this.
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Sadly I don't think there are any useful sublemmas. I shortened the proof a little.
jcommelin
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May 15, 2020
…prover-community#1554) * feat(field_theory/finite): cardinality of images of polynomials * docstrings * Johan's suggestions * slightly shorten proof
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One lemma about the cardinality of images of polynomials of roots of polynomials, and another about existence of roots of polynomials of a particular form over finite fields of odd cardinality.
These lemmas were used in a proof of the four squares theorem (every natural number is the sum of four squares, on Freek's list). They're difficult lemmas for the library, because they're not really textbook theorems, or theorems you might expect to find, but they still might be useful.
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