Today we reviewed for the final.
6/7
6/5
We defined the "algebraic closure" of one field inside another. We also defined what it means for a field to be "algebraically closed". Our main example was the field of complex numbers. At the end of class we stated the big theorem that algebraically closed extensions exist for every field.
6/3
Today talked about algebraic extensions. We proved that all finite extensions are algebraic using linear algebra. We also introduced the notion of the degree of an extension, which is another word for dimension. The degree is a useful tool in studying field extensions.
5/31
You should have continued talking about vector spaces with the sub.
5/29
Today there was a sub and you should have talked about vector spaces more.
5/27
No class today because of a holiday.
5/24
Lecture today was spent reviewing vector spaces and the fundamentals of linear algebra. This should be familiar for anyone who took a course in linear algebra. The main difference is that our vector spaces are defined over an arbitrary field. There will be some homework problems which cover the usual rank-nullity theorem, linear transformations, and other common topics. Reminder: I will be gone next week and there will be a sub.
5/22
Today we defined simple extensions, which are of the form F(α) for some α. These are the smallest fields containing F and α. We showed that every element in a simple algebra extension F(α)/F can be written as a polynomial in α of degree n-1, where n is the degree of α over F.
5/20
We defined algebraic and transcendental elements in extension fields. An element is algebraic if it satisfies a polynomial with coefficients in the ground field. Otherwise, it is transcedental. π is an example of a transcental element, √(2) is an example of an algebraic element.
5/17
Today we went proved that given any field F and polynomial f in F[x], there is a field E and element α in E such that f(α) = 0. This is important because it shows that given a polynomial, there is always a bigger field which contains a zero. Moreover, we have a way to construct such a field via quotient rings of F[x].
5/15
We defined prime ideals today and showed that an ideal is prime if and only if the quotient is an integral domain. This was similar to the theorem we had last lecture about maximal ideals and fields. At the end of class, we looked at maximal ideals in polynomial rings over a field. We shows that all ideals in F[x] are principle, and if p(x) is irreducible, then the ideal (p) is maximal.
5/13
Today we continued talking about ideals. We defined maximal ideals and proved that an ideal is maximal if and only if the quotient is a field (under the hypothesis that the ring is commutative with unity).
5/10
Today was the midterm. It should be graded and returned by sometime next week.
5/8
Today we reviewed for the midterm on Friday. We spent class mostly answering questions people had on homework, lecture, the book, quizzes, and the midterm review. Good luck on the exam!
5/6
We defined nilpotent elements and the nilradical of a ring. We did a few examples such as showing that 2 in ℤ/4ℤ is nilpotent. We also talked about the exam and how important it is to write clearly and legibly.
5/3
Today we introduced ideals. These are important objects. They allow us to construct quotient rings. A very important example of an ideal is the kernel of a ring homomorphism. We summaries one of the "isomorphism theorems" which states if if Φ:R → R' is a ring homomorphism, then it induces an isomorphism if R/N → R'.
5/1
We proved unique factorization of polynomials into products of irreducibles. The uniqueness was only up to order and constant factors. At the end of class, we talked about ring homomorphisms today. We went over several properties such as, if Φ:R → R', then Φ(R) is a subring of R'.
4/29
Today we focused on polynomials over ℚ[x] and ℤ[x]. The main theorem was that if a polynomial in ℤ[x] factors over ℚ[x], then it also factors over ℤ[x]. We used this and some other corollaries to prove that the pth cyclotomic polynomial Φp is irreducible.
4/26
We continued talking about polynomials today. We proved that f(a)=0 if and only if x-a divides f(x). We used this to prove that any finite subgroup of the multiplicative group of a field must be cyclic. Towards the end of class, we introduced irreducible polynomials and looked at the relation between irreducibility and having no zeros.
4/24
Today we proved the division algorithm in rings of the form F[x] for a field F. This says that given polynomials f(x) and g(x) with deg(g) > 0, there exists unique polynomials q(x) and r(x) such that f = qg + r and deg(r) < deg(g). We also practiced finding q and r via polynomial long division.
4/22
We spent most of today working on a selection of problems from the book which were not assigned as homework. This included 22.27, 22.28, 22.29, and 22.30. We will go over solutions to any problems people have questions about next class.
4/17
Today was a slight digression into some history. We saw how our number system has evolved over time because of the need to solve certain polynomial equations. Toward the end of class we worked out several example problems including one asking for rings where a polynomial had many zeros.
4/15
Today we discussed the "field of fractions" or "field of quotients". This is a construction possible for any integral domain D. It is the smallest field containing D. The prototypical example of a field of frations if the field ℚ over the integers ℤ.
4/12
In lecture we showed that the units of a ring form a group. We then focused on the units in ℤ/nℤ and showed that they are exactly the a such that gcd(a,n) = 1. Then we defined the Euler-phi function φ(n), which counts the number of positive integers less than n which are coprime with n.
4/10
Today we defined the characteristic of a ring and then moved on to Fermat's Little Theorem (FLT). We used FLT to prove a few things such as 11 does not divide 2^(11213)-1 and n^(33) - n = 0 modulo 15. The tricks we used in the proof were all versions of FLT.
4/8
We introduced "integral domains". These are commutative rings with unity (different from 0) that have no 0 divisors. Integral domains are an important class of rings we will continue to study. At the end of class, we proved with a counting based argument that every finite integral domain is a field.
4/5
Today we covered a lot of new terminology: division rings, commutative rings, rings with unity, subrings, and fields. We will be talking about fields for much of the later parts of our course. Make sure you feel comfortable with all of these definitions and can distinguish between them all.
4/3
Today we continued talking about rings and went through several more examples, including products of rings. We also proved several properties about rings such as 0*a = 0 for any ring element a. Here 0 is the addititive identity of the ring. We also introduced ring homomorphisms, which are maps that preserve both the additive and multiplicative structure.
4/1
Welcome to Math 120B! In this course we will learn about rings, ideals, fields, UFDs, PIDs, homomorphisms, isomorphisms, and more.