Extension fields and automorphisms are the most amazing thing I have ever learned. Essentially, when you think of them together, they are like a flower whose petals communicate with each other, but the center remains fixed.

 

I’m still reading the part on extension fields and automorphisms. But below are the thoughts that are more established and mature…

• Though in algebra textbooks it is always arranged as groups, rings, fields, groups and field actually appear more naturally than rings. The former is closed under addition and substraction. The latter is closed under all four arithmetic binary operations. Rings are an arbitrary stop in between that seem to stem from number theoretical considerations. Therefore, if there are other civilizations, I believe that they will define groups and fields the same way as we do, but they may not see rings as something fundamental. Rather, they may well think integral domains as fundamental and call our ring “a half integral domain”.

 

• Algebra iswas the study of polynomials. But today it is the study of “structures”, i.e. special sets with special operations. This is not only a change of language from equations to set- and map-theoretic expressions, but also a revolution that unveil hidden symmetry and patterns.

 

• It is an important change of perspective (Perspektivwechsel) that in modern algebra, we don’t see polynomials as a bunch of messy multiplications and additions, but as a vector space whose basis elements are powers of the indeterminate x. This allows linear algebra, i.e. the idea of dimensions, to kick in. Our good old linear algebra appears very naturally here. For example, a simple extension of a field F with an “external” element α algebraic over F of degree n is essentially a vector space whose basis is {1, α, α square, …,  α to the power of n-1}. Then it follows immediately from the invariance of the dimension that any other element β in F(α) will be algebraic to F, and with degree at most n.

 

• Another important change of perspective is that instead of talking about roots of polynomials, we talk in the language of the evaluation homomorphism on the polynomial ring. A root of a polynomial is not a number anymore, but is actually itself a function that maps the polynomial to zero. The set of all polynomials that are mapped to zero is then naturally the kernel of the evaluation homomorphism.

 

• We can’t overemphasize the concept of ideals. They come from “modules” in number theory. The fact that we can classify integers according to their remainders with respect to the module implies that we can also classify polynomials according to with respect to something similar, such that elements of the same class behave very similarly, i.e. we can choose any representative of a class to do additions and multiplications. There will be a class that behaves just so much like zero such that the result of anything multiplied by it is still it. This is our ideal. The change of modules to ideals is our third important change of perspective. The kernel of a homomorphism is by birth an ideal.

 

• Precisely because of ideals, we are able to construct a root to any random polynomial by brutal force. Where the root is is another story… 🙂 (This is Kronecker’s theorem.)

 

• Also using the tool of ideals, we are able to prove the other amazing result that an algebraic number over a field always corresponds uniquely to a monic irreducible polynomial. This result, in turn, proves that the set of all algebraic numbers has the same number of elements to the set of all integers.

 

• An algebraic number corresponds uniquely to a monic irreducible polynomial, but the converse is not true. A irreducible polynomial may correspond to multiple algebraic numbers, and they are called conjugates. For example, -sqrt(2) and sqrt(2) both correspond to x^2-2 in Q[x], so they are conjugates.

 

• The mapping of an algebraic number to its conjugates is probably why we are studying automorphisms. Again, I’m still reading this part.

Be right back 🙂