Simple proof of Chebotarev's theorem
Péter E. Frenkel · arXiv (Cornell University) · 2003
We give a simple proof of Chebotarëv’s theorem: Let p be a prime and ω a are non-zero. primitive pth root of unity. Then all minors of the matrix ( ω ij) p−1 i,j=0 Let p be a prime and ω a primitive pth root of unity. We write Fp for the field with p elements. In 1926, Chebotarëv proved the following theorem (see [3]): Theorem. For any sets I, J ⊆ Fp with equal cardinality, the matrix (ωij)i∈I,j∈J has non-zero determinant. Independent proofs were given by Dieudonné [1], Evans and Isaacs [2], and Terence Tao [4]. Tao points out that the theorem is equivalent to the inequality |suppf|+|supp ˆ f | ≥ p+1 holding for any function 0 ̸ ≡ f: Fp → C and its Fourier transform ˆ f, a fact also discovered independently by András Biró. We give a very simple proof via the following two lemmas. Lemma 1 Let Ω be an indeterminate. Then we have a commutative diagram