| Lectures and Topics by date
- M 1/4 Rings, ring of polynomials, basic properties of ring operations, subrings, ring homomorphisms.
Here is the link to the first lecture.
Because this is the first lecture, I have made it shorter than usual.
- W 1/6: Ring isomorphism, subring criterion, image and kernel of ring homomorphisms,
ring homomorphisms from \(\mathbb{Z}\) to a unital ring, the evaluation or substitution map.
Here is the link to the second lecture.
- F 1/8: The evaluation or the substitution homomorphisms, units and fields, zero-divisors and integral domains, characteristic
of a unital ring.
Here is the link to the third lecture.
- M 1/11: Field of fractions.
Here is the link to the forth lecture.
- W 1/13: Field of fractions, ideals, quotient rings, and the first isomorphism.
Here is the link to the fifth lecture.
- F 1/15: Evaluation map and the first isomorphism, degree of polynomials, \(D\) is an integral domain implies
\(D[x]\) is an integral domain, and the most general form of long division for polynomials.
Here is the link to the sixth lecture.
- W 1/20: The factor theorem, the generalized factor theorem, \(x^p-x=x(x-1)\cdots (x-(p-1))\) in \(\mathbb{Z}_p[x]\), Wilson's theorem,
\(F[x]\) is a PID if \(F\) is a field, Euclidean domain, Euclidean domain implies PID.
Here is the link to the seventh lecture.
- F 1/22: The ring of Gaussian integers is a ED and a PID, algebraic numbers, minimal polynomials,
characterization of minimal polynomials, elements of quotient rings of ring of polynomials.
Here is the link to the eighth lecture.
- M 1/25: Review of materials.
- W 1/27: Assuming that \(\alpha \) is algebraic over a field \(F\), every element of \(F[\alpha]\) can be uniquely written
as an \(F\)-linear combination of \(1,\alpha,\ldots, \alpha^{n-1}\) where \(n\) is the degree of the minimal polynomial of \(\alpha\) over \(F\).
Irreducible elements of integral domains. In an integral domain, \(\langle a\rangle=\langle b\rangle\) if and only if \(a=bu\) for some unit \(u\).
In a unital commutative ring \(\langle a\rangle\) is not proper if and only if \(a\) is a unit. A unital commutative ring is a field if and only if
it has exactly two ideals. \(a\in D\) is irreducible if and only if \(\langle a\rangle\) is maximal among proper pricipal ideals of \(D\). Maximal ideals.
Ideals of \(A/I\) are of the form \(J/I\) where \(J\) is an ideal of \(A\) which contains \(I\). In a unital commutative ring \(A\) an
ideal \(I\) is maximal if and only if \(A/I\) is a field. In a PID \(D\) which is not a field, \(a\) is irreducible if and only if \(D/\langle a\rangle\)
is a field. If \(E\) is a field extension of \(F\) and \(\alpha\in E\) is algebraic over \(F\), \(F[\alpha]\) is a field.
Here is the link to the ninth lecture.
- F 1/29: Zeros and irreducibility of polynomials. Degree 2 and 3 irreducibility criterion. A field of order 27. The rational
root criterion. A monic integer polynomial does not have a rational zero if it does not have a zero in \(\mathbb{Z}_n\). How this can help us in
conjunction with the Fermat's little theorem.
Here is the link to the tenth lecture.
- M 2/1: Content of non-zero polynomials in \(\mathbb{Q}[x]\). Gauss's lemma. A non-constant primitive polynomial is irreducibile in \(\mathbb{Q}[x]\)
if and only if it is irreducible in \(\mathbb{Z}[x]\). Mod-\(p\) irreducibility criterion.
Here is the link to the eleventh lecture.
- W 2/3: How to use the mod-\(p\) irreducibility criterion. Eisenstein's irreducibility criterion. Some applications of Eisenstein's irreducibility criterion.
What a Unique Factorization Domain is. Writing elements as a product of irreducible elements and the ascending chain condition. A ring is Noetherian if and only if all of
its ideals are finitely generated. Every PID is Noetherian, and in a PID every non-zero non-unit element can be written as a product of irreducible elements.
Here is the link to the twelfth lecture.
- F 2/5: Review of materials
- M 2/8: For an integral domain \(D\) where every non-zero non-unit can be written as a product of irreducible elements,
\(D\) is a UFD if and only if every irreducible is prime. In every integral domain, primes are irreducible. PID implies UFD, and so \(\mathbb{Z}\), \(F[x]\) where
\(F\) is a field, \(\mathbb{Z}[i]\), and \(\mathbb{Z}[\omega]\) where \(\omega=\frac{-1+\sqrt{-3}}{2}\) are UFDs. An ideal \(I\) of a unital commutative ring \(A\) is
prime if and only if \(A/I\) is an integral domain.
Here is the link to the thirteenth lecture.
- W 2/10: The main result that we proved was \(\mathbb{Z}[x]\) is a UFD. Along the way we showed: For \(c\in \mathbb{Z}\) we have \(c\) is irreducible in
\(\mathbb{Z}\) if and only if \(c\) is irreducible in \(\mathbb{Z}[x]\), and \(c\) is prime in
\(\mathbb{Z}\) if and only if \(c\) is prime in \(\mathbb{Z}[x]\). Then we consider the function
\({\rm prim}:\mathbb{Q}[x]\setminus\{0\} \rightarrow \mathbb{Z}[x], {\rm prim}(f)=\bar{f}\) where \(\bar{f}\) is the primitive form of \(f\). We proved that
- \(f\) is a unit in \(\mathbb{Q}[x]\) if and only if \(\bar{f}\) is a unit in \(\mathbb{Z}[x]\).
- \(f|g\) in \(\mathbb{Q}[x]\) if and only if \(\bar{f}|\bar{g}\) in \(\mathbb{Z}[x]\).
- \(f\) is irreducible in \(\mathbb{Q}[x]\) if and only if \(\bar{f}\) is irreducible in \(\mathbb{Z}[x]\).
- \(f\) is prime in \(\mathbb{Q}[x]\) if and only if \(\bar{f}\) prime in \(\mathbb{Z}[x]\).
Here is the link to the fourteenth lecture.
- F 2/12: For a UFD \(D\) and an irreducible \(p\) we defined the \(p\)-valuation. We defined the greatest common divisor of a finite set of elements of
a UFD. We defined the content of a non-zero polynomial in \(Q(D)[x]\) where \(Q(D)\) is the field of fractions of a UFD. Guass's lemma for UFDs was proved. One can
use these tools to prove that \(D[x]\) is a UFD if \(D\) is a UFD. Then inductively one can deduce that \(D[x_1,\ldots,x_n]\) is a UFD if \(D\) is a UFD. In
particular, \(\mathbb{Z}[x_1,\ldots,x_n]\) and \(F[x_1,\ldots,x_n]\), where \(F\) is a field, are UFDs.
Here is the link to the fifteenth lecture.
- F 2/19: We prove the existence of splitting fields, and work towards the uniqueness of splitting fields.
Here is the link to the sixteenth lecture.
- M 2/21: We prove the uniqueness of splitting fields, and mentione a couple of examples.
Here is the link to the seventeenth lecture.
- W 2/23: We proved the existence and the uniqueness, up to an isomorphism, of finite fields of order \(p^n\). We proved that a finite field of order \(p^n\) is a
splitting field of \(x^{p^n}-x\) over \(\mathbb{Z}_p\). \(\mathbb{F}_{p^n}\) denotes a finite field of order \(p^n\). We proved that \(x^{q}-x=\prod_{\alpha\in \mathbb{F}_q} (x-\alpha)\).
The derivative of a polynomial is defined. We proved that if \(F\) is a field and \(f\in F[x]\), then \(f\) does not have multiple zeros in its splitting field exactly when \(\gcd(f,f')=1\)
in \(F[x]\).
Here is the link to the eighteenth lecture.
- F 2/25: Vector spaces over an arbitrary field were discussed. Proved that the cardinality of a spanning set is at least the cardinaltiy of an independent set. Proved that
every spanning set contains a basis as a subset. Proved that every two bases have the same cardinality. Defined the quotient of a vector space by one of its subspaces. Proved that
\(\dim W +\dim (V/W)=\dim V \). Proved the first isomorphism theorem of vector spaces. Proved the kernel-image theorem.
Here is the link to the nineteenth lecture.
- M 3/1: \(\dim_F F[x]/\langle f\rangle=\deg f\). Degree of a field extension. \([F[\alpha]:F]=\deg m_{\alpha,F}\) if \(\alpha\) is algebraic over \(F\).
If \(\mathbb{F}_{p^m}\) can be embedded into \(\mathbb{F}_{p^n}\), then \(m|n\). The Tower Rule: \([L:F]=[L:E][E:F]\). Finite extensions are algebraic.
The algebraic closure of a field in a given field extension. Construction problems in Euclidean geometry. \(\sqrt[3]{2}\) cannot be constructed using ruler and compass.
If \([E:\mathbb{Q}]\) is a power of 2, then \(x^3-2\) is irreducible in \(E[x]\). If \([F[\alpha]:F]\) is odd, then \(F[\alpha]=F[\alpha^2]\), and some other applications of the tower rule.
Here is the link to the twentieth lecture.
- F 3/5: The \(n\)-th cyclotomic polynomial \(\Phi_n(x)\) is defined. We prove that \(\Phi_n(x)\) is an integer polynomial which is irreducible in \(\mathbb{Q}[x]\).
We deduce that \(m_{\zeta_n,\mathbb{Q}}(x)=\Phi_n(x)\) and \([\mathbb{Q}[\zeta_n]:\mathbb{Q}]=\phi(n)\).
Here is the link to the twenty first lecture.
- M 3/8: It is proved that if \(E\) is a field extension of \(F\) and \([E:F]< \infty \), then the following are equivalent.
- For some \(f\in F[x]\), \(E\) is a splitting field of \(f\) over \(F\).
- For every field extension \(L\) of \(E\) and every \(\theta\in {\rm Aut}_F(L)\), we have \(\theta(E)=E\).
- For every \(\beta\in E\), there are \(\beta_i\)'s in \(E\) such that \(m_{\beta,F}(x)=\prod_i (x-\beta_i)\).
We say \(E\) is a normal field extension of \(F\) if the third property mentioned above holds.
Here is the link to the twenty second lecture.
- W 3/10:
Suppose \(E\) is a finite normal extension of \(F\). Then \(r_{L,E}:{\rm Aut}_F(L)\rightarrow {\rm Aut}_F(E), r_{L,E}(\theta):=\theta|_E\)
is a well-defined group homomorphism and its kernel is equal to \({\rm Aut}_E(L)\). In particular, \({\rm Aut}_E(L)\) is normal subgroup of \({\rm Aut}_F(L)\) if
\(E\) is a finite normal extension of \(F\). If in addition \(L\) is a normal extension of \(F\), then \(r_{L,E}\) is surjective, and by the first isomorphism of groups,
we have that \({\rm Aut}_F(L)/{\rm Aut}_E(L)\simeq {\rm Aut}_F(E)\).
We discussed how the normal extension property behaves in a tower of field extensions. We showed that if \(F\subseteq E\subseteq L\) is a tower of field extensions and
\(L\) is a normal extension of \(F\), then \(L\) is a normal extension of \(F\). We provided examples where
- \(L\) is a normal extension of \(F\), but \(E\) is not a normal
extension of \(F\), and
- \(L\) is a normal extension of \(E\) and \(E\) is a normal extension of \(F\), but \(L\) is not a normal extension of \(F\).
We discussed what the normal closure of a field extension is, and proved its existence.
Prove that if \(E\) is a finite normal extension of \(F\), then \(|{\rm Aut}_F(E)|\leq [E:F]\). Proved that if \(f\in F[x]\), \(E\) is a splitting field of \(f\) over
\(F\), and all of irreducible factors of \(f\) have distinct zeros in \(E\), then \(|{\rm Aut}_F(E)|= [E:F]\).
Here is the link to the twenty third lecture.
- F 3/10: We defined separable algebraic field extensions, and proved that for a finite field extension \(E\) of \(F\), the following statements are equivalent:
- \(E\) is a normal separable extension of \(F\).
- \(E\) is a splitting field of a separable polynomial \(f\in F[x]\) over \(F\).
- \(|{\rm Aut}_F(E)|=[E:F]\).
We say an algebraic field extension \(E\) of \(F\) is a Galois extension if it is both normal and separable. We will explore Galois Theory in 100C.
Here is the link to the twenty fourth lecture.
|
ucsd 
edu