Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 38 (1997), No. 2, 213-215.

A simple, (almost) non-inductive Proof for the Existence of Composite Fields and Algebraic Closures

Andreas W. M. Dress

FSMP - Strukturbildungsprozesse, Universität Bielefeld Postfach 10 01 31, 33501 Bielefeld e-mail: jordan@mathematik.uni-bielefeld.de

In the theory of fields, it is often convenient to know that for any two extension fields L_{1} and L_{2} of a common ground field K, there exists a field extension L of K together with two embeddings φ_{1} : L_{1} ↪ L and φ_{2} : L_{2} ↪ L with φ_{1}|_{K} = φ_{2}|_{K} = Id_{K}. In addition, L may be assumed to be generated by its subfields φ_{1}(L_{1}) and φ_{2}(L_{2}) in which case L is often called a composite field extension of K – that is, a field extension composed from L_{1} and L_{2} (over K). Standard proofs of this fact either use tensor products (a rather time consuming concept for an undergraduate lecture) or construct L by inductively adjoining to L_{2} elements x_{1}, x_{2}, … from L_{1} with L_{1} = K(x_{1}, x_{2}, …) (a rather unpleasant procedure, in particular if L_{1} needs an infinite number of generators). Hence, it might be worthwhile to observe that – provided one is willing to use the existence of maximal ideals M in any ring R (with 1_{R} ∈ R and 1_{R} ≠ 0_{R}!) – a much simpler, non-inductive (except for the use of the existence of maximal ideals), and much less time consuming way is to consider the K-algebra R′ := End_{K}(V) of K-endomorphisms of the K-vectorspace V := Hom_{K}(L_{1}, L_{2}) into which L_{1} can be embedded K-linearly by

Φ_{1} : L_{1} → R′ : x_{1} ↦ (V → V : ψ ↦ (ψ^{x_{1}} : L_{1} → L_{2} : y_{1} ↦ ψ(y_{1} · x_{1})))

while L_{2} can be embedded K-linearly into R′ via

Φ_{2} : L_{2} → R : x_{2} ↦ (V → V : ψ ↦ (^{x_{2}}ψ : L_{1} → L_{2} : y_{1} ↦ x_{2} · ψ(y_{1}))).

As Φ_{1}(x_{1}) ∘ Φ_{2}(x_{2}) = Φ_{2}(x_{2}) ∘ Φ_{1}(x_{1}) holds for all x_{1} ∈ L_{1} and x_{2} ∈ L_{2}, the subring R of R′ generated by Φ_{1}(L_{1}) and Φ_{2}(L_{2}) is commutative. In addition, Φ_{1}|_{K} = Φ_{2}|_{K} = Id_{K}

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1997 Heldermann Verlag

clearly holds by the definition of K-linearity (relative to the canonical identification of the elements of K with the associated scalar multiplication in V considered as an element in R). And V ≠ {0} implies 1_{R} ≠ 0_{R}. So, choosing a maximal ideal M in R, putting L := R/M, φ_{1} : L_{1} → L : x_{1} ↦ Φ_{1}(x_{1}) + M, and φ_{2} : L_{2} → L : x_{2} ↦ Φ_{2}(x_{2}) + M, the existence of a composite field extension L of K for given field extensions K ⊆ L_{1} and K ⊆ L_{2} is easily established.

To get a more symmetric version of this proof, we may replace V = Hom_{K}(L_{1}, L_{2}) by the K-vectorspace B(L_{1} × L_{2}, K) of all K-bilinear maps from L_{1} and L_{2} into K and R′ by the ring of its K-endomorphisms into which L_{1} and L_{2} embed in a similar fashion. The advantage of this approach is that it generalizes immediately to an arbitrary number of field extensions: let I be an arbitrary set and, for each i ∈ I, let L_{i} denote some field extension of a common ground field K. Let

V := Mult(∏_{i∈I} L_{i}, K) := {f : ∏_{i∈I} L_{i} → K | for each i_{0} ∈ I

and any choice of elements x_{j} ∈ L_{j} (j ∈ I − {i_{0}}),

the induced map f : L_{i_{0}} → K : x_{i_{0}} ↦ f((x_{i})_{i∈I}) is K-linear}

denote the K-vectorspace of all multilinear maps from ∏_{i∈I} L_{i} into K and let R′ := End_{k}(V)

denote the K-algebra of its K-endomorphisms. Clearly, we have 1_{R′} ≠ 0_{R′} in view of V ≠ {0} which in turn is an immediate consequence of the fact that for any choice of surjective Klinear maps λ_{i} : L_{i} ↠ K(i ∈ I), the map

f_{(λ_{i})_{i∈I}} : ∏_{i∈I} L_{i} → K : (x_{i})_{i∈I} ↦ \begin{cases}∏_{i∈I} λ_{i}(x_{i}) & if λ_{i}(x_{i}) = 1 for almost all i ∈ I, \\ 0 & else \end{cases}

is a non-zero element in V. Hence, as above, there exist canonical embeddings

Φ_{i_{0}} : L_{i_{0}} → R′ : x_{i_{0}} ↦ (V → V : f ↦ (∏_{i∈I} L_{i} → K : (x_{i})_{i∈I} ↦ f(y_{i})

with y_{i} := \begin{cases}x_{i} & if i ≠ i_{0} \\ x_{i_{0}}x_{i} & else \end{cases} )) ,

the subring R of R′ generated by all the images Φ_{i}(L_{i}) (i ∈ I) of the L_{i} is commutative and satisfies the condition 1_{R} ≠ 0_{R} and, for all i, j ∈ I, one has Φ_{i}|_{K} = Φ_{j}|_{K} = Id_{K}. Hence, choosing a maximal ideal M in R as above, the existence of a composite field extension L of K for any arbitrarily large family of field extensions L_{i} (i ∈ I) of K is established.

Choosing for I the set of all irreducible (or just all non-constant) polynomials in K[x] and for i = i(x) ∈ I the splitting field of i(x) as the associated field extension L_{i} of K, it is easily seen that this way an algebraically closed field extension L of K will result, thus establishing in a rather simple and straightforward fashion the basic fact that any field K is a subfield of an algebraically closed field.

It is also obvious that the same approach yields, for any family of simple associative K-algebras A_{i} (i ∈ I), the existence of a simple associative K-algebra A together with Kalgebra embeddings φ_{i} : A_{i} → A (i ∈ I) such that A is generated by ⋃_{i∈I} φ_{i}(A_{i}) and the

identity φ_{i}(a_{i}) · φ_{j}(a_{j}) = φ_{j}(a_{j})φ_{i}(a_{i}) holds for all i, j ∈ I with i ≠ j and all elements a_{i} ∈ A_{i} and a_{j} ∈ A_{j}.

It is also easy to derive, either by the same method or simply as a consequence of the above result, the following slightly more general result needed in the analysis of normal field extensions:

Given an arbitrary index set I, a family K_{i} ↪ L_{i} (i ∈ I) of field extensions and embeddings α_{i} : K_{i} ↪ K (i ∈ I) of the various ground fields K_{i} into a common “super” ground field K, there exists a field extension ψ : K ↪ L of K and embeddings φ_{i} : L_{i} ↪ L (i ∈ I) with φ_{i} |_{K_{i}}= ψ ∘ α_{i} for all i ∈ I:

Indeed, first choosing composite field extensions K_{i} ↪ L′_{i} of K_{i} ↪ L_{i} and α_{i} : K_{i} ↪ K we get field extensions ψ_{i} : K ↪ L′_{i} with embeddings β_{i} : L_{i} ↪ L′_{i} satisfying the relation β_{i} |_{K_{i}}= ψ_{i} ∘ α_{i} for each i ∈ I individually, which then can be composed to one big field extension ψ : K ↪ L with embeddings φ′_{i} : L′_{i} ↪ L satisfying the relation φ′_{i} ∘ ψ_{i} = ψ and, hence, with embedding φ_{i} := φ′_{i} ∘ β_{i} : L_{i} ↪ L satisfying the relation

φ_{i} |_{K_{i}}= φ′_{i} ∘ β_{i} |_{K_{i}}= φ′_{i} ∘ ψ_{i} ∘ α_{i} = ψ ∘ α_{i}

for each i ∈ I, as claimed.

Similarly, one can go directly for this more general result by first considering the Kvectorspace V of all maps f : ∏_{i∈I} L_{i} → K which for each i_{0} ∈ I and all x_{j} ∈ L_{j} (j ∈ I \ {i_{0}})

induce a K_{i_{0}}-linear map L_{i_{0}} → K : x_{i_{0}} ↦ f((x_{i})_{i∈I}) relative to the K_{i_{0}}-vectorspace structure of L induced by α_{i_{0}} : K_{i_{0}} → L and observing that, as above, one has V ≠ {0}, and then embedding the L_{i} into the K-endomorphism ring R of V and further arguing as above.

And finally, also as above, there is a non-commutative version of this result, relating to simple K_{i}-algebras A_{i} giving rise to a simple K-algebra A relative to field extensions α_{i} : K_{i} ↪ K or – only seemingly more generally – embeddings α_{i} : K_{i} ↪ K of the various ground fields K_{i} into the centre of a simple algebra K.

Received May 6, 1996