Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 5th 2015

    Created splitting field.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 6th 2015

    Made improvements to splitting field, noting in particular that existence and uniqueness up to isomorphism of splitting fields of arbitrary sets of polynomials doesn’t require the full axiom of choice, but only the ultrafilter principle. This applies in particular to algebraic closures.

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 6th 2015

    Is it worth pointing out that for a countable field (and so a countable set SS of polynomials) one doesn’t need such a strong axiom to get algebraic closure etc? Or better, that for a set of polynomials with bounded cardinality, there much be an ultrafilter principle (for sets up to a certain size) that implies the existence of splitting fields.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 6th 2015

    Sure, there are a number of such statements one could make: for finite fields no choice is required, and for any field (such as \mathbb{Q}) which sits canonically inside \mathbb{C} you don’t need choice, in addition to the statements you gave. Please feel free to add such, or I might do so sometime later.

    • CommentRowNumber5.
    • CommentAuthorZhen Lin
    • CommentTimeJul 6th 2015

    Isn’t there some subtlety even in the countable case? I vaguely recall something terrible like, a countable field has a unique countable algebraic closure, but there may also be uncountable algebraic closures.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 6th 2015
    • (edited Jul 6th 2015)

    Right, there was this thread on FOM started by Timothy Chow about uniqueness of algebraic closure of even \mathbb{Q} if one doesn’t assume some amount of choice or choice-consequence, perhaps such as the statement that the union of countably many finite sets is countable. I haven’t looked at the cited paper, but it sounds like Lauchli has constructed these monsters that Zhen Lin is referring to. I didn’t trace through to the end of the discussion there, which proceeds by fits and (false) starts. (Sorting by thread is highly recommended.)

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 7th 2015
    • (edited Jul 7th 2015)

    Let me point out for completeness that the thread continues, at times, in the following month. No conclusion seems to be reached!

    Note that the contention is mostly around the definition of algebraic closure.