Replies

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  Jakobjdw@mathstodon.xyz

  Oct 27, 2026, 8:20 AM

  @gthm Let's forget about the generic point for a moment, and let's assume that everything is separable (for example in char. 0). Let K be the algebraic closure of k. There is a natural map K -> A^1_k mapping an element z \in K to the ideal of polynomials vanishing on z. This is a prime ideal, so it is a point in A^1_k.

  The map is surjective onto the closed points of A^1_k: Every maximal ideal in k[x] is generated by an irreducible polynomial f which has a root z \in K. This z is a preimage.

  Two elements in K are identified under the map (i.e. they have the same minimal polynomial) if and only if they are conjugate under a Galois automorphism. This is a standard result from Galois theory. The non-trivial direction is that elements with the same minimal polynomial f are conjugate. In a first step it is enough to show that they are conjugate in the splitting field of f, this result is typically formulated as »the Galois group of a polynomial acts transitively on its roots« in introductions to Galois theory. The automorphism can then be extended to the algebraic closure.

  Hence the map induces a bijection between points of K modulo the Galois group onto the set of closed points of A^1_k. The only additional point is the generic point.

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  Jakobjdw@mathstodon.xyz

  Oct 27, 2026, 8:26 AM

  @gthm For example if k = R and K = C, both elements i and -i of C are mapped to the prime ideal generated by (x^2 + 1). The element 5 is mapped to the prime ideal (x - 5). In general, two elements of C have the same minimal polynomial over R if and only if they are equal or one is the complex conjugate of the other. We have A^1_R = C / (complex conjugation) + generic point.

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  ⏚ Antoine Chambert-Loirantoinechambertloir@mathstodon.xyz

  Oct 27, 2026, 8:56 AM

  @gthm @jdw you didn't really need Galois theory: just identify two numbers, say √2 and -√2, when they are solutions of the same polynomial equation, here T²-2=0, which cannot be written as the product of two equations of smaller degrees.

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