Last lecture we characterized subfields and used the characterization to provide many new examples of fields. Now we start to explore systematically which subfields of the complex numbers are suitable to study the question of which polynomial equations can be solved.
Definition 1 If is a field and is a subfield, we write and also say that is an extension of
Definition 2 If is a field, and we denote by the smallest subfield of that contains both and If we write We say that is the subfield of generated by over
For example, last lecture we showed that
“Smallest” in the definition above refers to containment. Put another way, we are claiming that there is a field that contains and and such that given any subfield of that also contains and then This requires an argument.
Theorem 3 Let be a field. Let be a nonempty collection of subfields of Then is a field.
Proof: We use the characterization from last lecture to show that is a subfield of It is straightforward to check that this set is closed under addition and multiplication (since all the fields in are), and that it contains additive and multiplicative inverses of all its (nonzero) elements. It also has at least two elements, since for any field
Corollary 4 If is a field, and then exists.
Proof: Let be the collection of subfields of that contain both and Note that since Then is a field, it contains both and and is contained in any subfield of that contains both and by definition of Therefore,
This is a characterization “from above.” As in the case of in many concrete examples it is possible to give a simple description (a characterization “from below”) of but this is in general a difficult problem.
Proof: We already know that By induction, for all since is closed under addition. Since is closed under additive inverses, Since is closed under multiplicative inverses (of nonzero elements) and under multiplication, then
Recall Gauß’ fundamental theorem of algebra, that we will take for granted in what follows. (That we take it for granted does not mean that this is an easy result. All its known proofs use deep facts about either about its algebraic structure or its analytic structure.)
Theorem 6 (Gauß) Let be a nonconstant polynomial with complex coefficients. Then has at least one complex root, i.e., there is some such that By induction, it follows that if has degree then it has exactly roots, taking into account their multiplicity.
Definition 7 Let be a subfield of and let be a nonconstant polynomial with coefficients in . Then denotes the smallest subfield of that contains and the roots of
We will usually use this notation with in place of In this case, thanks to Proposition 5, we have that is simply the smallest subfield of that contains the roots of
For example, if then
Clearly, This extension has several interesting properties. For example, it is a vector space over Next lecture we will study the important case when is an extension by radicals.
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