Field theory MOC

Algebraic closure

Let 𝐾 be a field. An algebraic closure ――𝐾 of 𝐾 is an algebraically closed field such that ――𝐾 :𝐾 is an algebraic extension. field Assuming AC,¹ an algebraic closure always exists and is unique up to isomorphism of field extensions, so one often speaks of the algebraic closure.

Proof of existence and uniqueness

The proof of existence and uniqueness requires enough lemmata to warrant a section of this Zettel.² We invoke Zorn’s lemma.

Existence

Let 𝐾 =𝐾0 be a field. There exists an extension 𝐾1 :𝐾0 such that every nonconstant polynomial 𝑓(𝑥) ∈𝐾[𝑥] has at least one root in 𝐾1.

Proof due to Emil Artin

Let F denote the set of nonconstant monic polynomials in 𝐾0[𝑥], and let 𝐾0[𝑡𝑓]𝑓∈F be the corresponding polynomial ring, potentially in infinitely many indeterminates. Consider the ideal

𝐼=⟨𝑓(𝑡𝑓):𝑓∈F⟩◃𝐾0[𝑥],

which we will show must be proper. Suppose towards contradiction that 𝐼 =⟨1⟩, so

1=𝑛∑𝑖=1𝑎𝑖𝑓𝑖(𝑡𝑓𝑖)

for some 𝑎𝑖 ∈𝐾0[𝑡𝑓]𝑓∈F and 𝑓𝑖 ∈F. We can then construct an extension 𝐹 :𝐾 where the polynomials (𝑓𝑖)𝑛𝑖=1 have roots (𝛼𝑖)𝑛𝑖=1, by iteratively Adjoining a root to a field. If we evaluate

1=𝑛∑𝑖=1𝑎𝑖𝑓𝑖(𝛼𝑖)=𝑛∑𝑖=1𝑎𝑖⋅0=0

which is a contradiction. Since 𝐼 is proper, invoking Zorn it is contained in a maximal ideal 𝔪, giving the the field extension

𝐾0[𝑡𝑓]𝑓∈F𝔪:=𝐾1:𝐾0,

where by construction every nonconstant monic (and thus nonconstant general) polynomial 𝑓(𝑥) has a root 𝜋(𝑡𝑓).

To guarantee the existence of all roots we iterate this process ad infinitum, so not only does 𝑓(𝑥) ∈𝐾0[𝑥] have a root 𝛼1 ∈𝐾1, but 𝑓(𝑥)/(𝑥 −𝛼1) ∈𝐾1[𝑥] has a root 𝛼2 ∈𝐾2, &c. This yields a chain of extensions

𝐾0↪𝐾1↪𝐾2↪⋯

Let 𝐿 be the union or limit of this chain. 𝐿 is an algebraically closed field, and [[Algebraic interior of a field extension#^p1|thus (𝐿 :𝐾)∘ is an algebraic closure of 𝐾]].

Proof

For every 𝑎,𝑏 ∈𝐿 we have 𝑎,𝑏 ∈𝐾𝑖 for some 𝑖 ∈ℕ0, so we can just work within whatever 𝐾𝑖 is necessary, since the result is independent. Thus 𝐿 is a field. If 𝑓(𝑥) ∈𝐿[𝑥], then 𝑓(𝑥) ∈𝐾𝑖[𝑥] for some 𝑖 ∈ℕ0, and thus it has a root in 𝐾𝑖+1 ≤𝐿.

Uniqueness

See Embedding an algebraic extension into an algebraically closed field.

Suppose ――𝐾1 and ――𝐾2 are algebraic closures of 𝐾. Then there exists an isomorphism of field extensions 𝜓 ∈𝖥𝗅𝖽𝐾(――𝐾1,――𝐾2).

Proof

By the above, there exists a homomorphism 𝑖 ∈𝖥𝗅𝖽𝐾(――𝐾2,――𝐾1), which is automatically injective (Field homomorphisms are injective). It is also surjective, by ^A3.

---

tidy| en | SemBr

Footnotes

1. Allegedly, existence follows from the weaker Compactness theorem for first order logic, see footnote 7 on 2009. Algebra: Chapter 0, p. 403 ↩

2. 2009. Algebra: Chapter 0, §VII.2.1, pp. 400–404 ↩