9.4. MORE ON ALGEBRAIC FIELD EXTENSIONS 211

Why use this more elaborate proof? I think it is because you can give other examplesof algebraically complete fields. For example, begin with Q and let the algebraic numbersbe denoted by A. These are those numbers which are roots of a polynomial having rationalcoefficients. Then consider A2 to be those complex numbers which are roots of a polyno-mial having coefficients in A. In general, let An be roots of polynomials with coefficientsin An−1. In general, if An−1 is countable, then so is An. This is routine to show using thefact that there are countably many polynomials of degree m for each m ∈ N. Each has atmost m roots. Thus A∞ ≡ ∪∞

n=1Anis countable because Q is. Now recall also that it wasshown that the algebraic numbers over a field are a field. Therefore, A∞is also a field be-cause any finite number of elements of A∞ must be in a single one of the fields An for largeenough n. Now consider Lemma 9.3.1 applied to a polynomial having real coefficients inA∞. These coefficients are in some An and so the root from C having these coefficientsis in An+1 ⊆ A∞. Now the rest of the argument goes similarly. You show using the sameconsiderations that every polynomial having real coefficients in A∞ has a root in A∞. Thenyou do the easy extension to the case where the coefficients in A∞ are complex. This fieldis clearly much smaller than C because it is countable, and yet it is algebraically complete.The standard analysis proof given earlier will obviously not work because it is based oncompactness considerations.

9.4 More on Algebraic Field ExtensionsThis is on field extensions. There are many linear algebra techniques which are used in thisdiscussion and it seems to me to be very interesting. I am following various algebra booksin assembling this material. I hope it is useful and that I have not diminished it too muchby my attempts to write it down, because it is clear to me that, even though it has nothing todo with my own interests, it is some of the most wonderful mathematics I have ever seen.

Consider the notion of splitting fields. It is desired to show that any two are isomor-phic, meaning that there exists a one to one and onto mapping from one to the other whichpreserves all the algebraic structure. To begin with, is a theorem about extending homo-morphisms. [26]

Definition 9.4.1 Suppose F, F̄ are two fields and that f : F→ F̄ is a homomorphism. Thismeans that

f (xy) = f (x) f (y) , f (x+ y) = f (x)+ f (y)

An isomorphism is a homomorphism which is one to one and onto. A monomorphism isa homomorphism which is one to one. An automorphism is an isomorphism of a singlefield. Sometimes people use the symbol ≃ to indicate something is an isomorphism. Thenif p(x) ∈ F [x] , say

p(x) =n

∑k=0

akxk,

p̄(x) will be the polynomial in F̄ [x] defined as

p̄(x)≡n

∑k=0

f (ak)xk.

Also consider f as a homomorphism of F [x] and F̄ [x] in the obvious way.

f (p(x)) = p̄(x)