 1/9: Recall prime and maximal ideals and how to identify them by looking at the quotient ring. Zorn's lemma. Show that union of a chain of ideals is an ideal. Recall what
a multiplicatively closed subset is. Show that, for every \(f\in A\), \(S_f:=\{1,f,f^2,\cdots\}\) is a multiplicatively closed set and for every prime ideal \(\mathfrak{p}\),
\(S_{\mathfrak{p}}:=A\setminus \mathfrak{p}\) is multiplicatively closed.
Start the proof of the following theorem: suppose \(\mathfrak{a}\) is an ideal of \(A\) and \(S\subseteq A\) is multiplicatively closed.
Suppose \(\mathfrak{a}\cap S=\varnothing\). letterspacing
\(
\Sigma_{\mathfrak{a},S}:=\{\mathfrak{b}\unlhd A\mid \mathfrak{a}\subseteq \mathfrak{b}, \mathfrak{b}\cap S=\varnothing\}.
\)
Then (1) \((\Sigma_{\mathfrak{a},S},\subseteq)\) has a maximal element, and (2) every maximal element of \(\Sigma_{\mathfrak{a},S}\) is a prime ideal.
Before proving this theorem, we mentioned two corollaries: (1) every proper ideal has a maximal divisor. (2) In the above setting, there exists a prime ideal
\(\mathfrak{p}\) such that \(\mathfrak{a}\subseteq \mathfrak{p}\) and \(\mathfrak{p}\cap S=\varnothing\).
 1/11: Compeleting the proof of the mentioned theorem about existence of prime divisors that do not divide elements of a
multiplicatively closed subset. Using this to show that the set of all the nilpotent elements of a ring is equal to the intersection of all the prime ideals.
Show that \(\frac{A[x]}{\mathfrak{a}[x]}\simeq (\frac{A}{\mathfrak{a}})[x]\), and use it to deduce that \(\mathfrak{p}[x]\) is a
prime ideal of \(A[x]\) if \(\mathfrak{p}\) is a prime ideal of \(A\). Show that \({\rm Nil}(A[x])={\rm Nil}(A)[x]\).
 1/13: defined Euclidean Domain. Showed that \(\mathbb{Z}\), \(F[x]\) where \(F\) is a field, and \(\mathbb{Z}[i]:=\{a+bi\mid a,b\in \mathbb{Z}\}\) are ED.
Prove that every ED is PID. Defined irreducible and prime elements, and being associates. Proved an ideal theoretic criteria for being irreducible, prime,
and being associates. Defined a UFD.
 1/20: Proved that in an integral domain every prime element is irreducible. Showed that if the existence part of being a UFD holds in an integral domain,
then it is UFD precisely when every irreducible is prime. Proved that in a PID an element is irreducible if and only if it is prime. Also showed that
\({\rm Spec}(D)=\{0\}\cup{\rm Max}(D)\) if \(D\) is a PID, and \({\rm Spec}(D)=\{0\}\) if and only if \(D\) is a field. Discussed that if in an integral domain \(D\) there exists a nonzero
element which cannot be written as a product of units and irreducibles, then \(D\) does not satisfy the ascending chain condition (acc). Defined a Noetherian ring.
 1/23: Proved that the following statements are equivalent: (1) \(A\) is Noetherian, (2) The acc holds in \(A\), (3) Every ideal of \(A\) is finitely generated.
Next, we turned our attention to properties of rings which can pass to the ring of polynomials. We started with a nonexample: we showed
\(A[x]\) is a PID if and only if \(A\) is a field. Next we proved an example: (Hilbert's basis theorem) \(A\) is Noetherian implies that \(A[x]\) is Noetherian.
 1/25: Started the proof of: \(D\) is a UFD implies \(D[x]\) is a UFD. To this end, first we proved that if \(D\) is an integeral domain, then the following statements
hold: (1) \(a\in D\) is irreducible if and only if \(a\in D[x]\) is irreducible and \(\deg a=0\).
(2) \(a\in D\) is prime if and only if \(a\in D[x]\) is prime and \(\deg a=0\). Next, we recalled the field of fractions of an integral domain, and its universal property.
Our goal is to use the fact that \(F[x]\) is a UFD if \(F\) is the field of fractions of \(D\). To that end, we defined the \(p\)valuation of a nonzero element in a UFD,
mentioned its main properties, and defined the gcd of nonzero elements of a UFD. Based on the gcd function, we defined the content of a polynomial and primitive polynomials.
We proved both versions of Gauss's lemma: (v1) product of two primitive polynomials is primitive; (v2) the content of product is the product of contents.
 1/27: First we proved: suppose \(D\) is a UFD and \(F\) is its field of fractions. Suppose \(f(x)\in D[x]\) and \(f(x)=\prod_{i=1}^n g_i(x)\) for some \(g_i(x)\in F[x]\).
Then there exist \(c_i\in F\) such that (1) \(\prod_{i=1}^n c_i=1\), and (2) \(\overline{g}_i(x):=c_ig_i(x)\in D[x]\). Used this to obtian many corollaries: suppose
\(f\in D[x]\) is primitive and \(\deg f>0\); then \(f\) is irreducible in \(D[x]\) if and only if it is irreducible in \(F[x]\). We proved that if \(f\in D[x]\) is primitive
and \(g(x)\in D[x]\), then \(fg\) in \(F[x]\) exactly when \(fg\) in \(D[x]\). Proved that \(D\) is a UFD implies that \(D[x]\) is a UFD. Proved that if \(f\in D[x]\)
is a monic polynomial and \(f \pmod{\mathfrak{p}}\) is irreducible for some prime ideal \(\mathfrak{p}\), then \(f\) is irreducible in \(F[x]\).
 1/30: Proved Eisentein's criterion; along the way, we defined the residue field of a prime ideal of a ring. Proved that \(\mathbb{C}[x,y]/\langle x^n+y^n1\rangle\) is a UFD. Proved that \(x^515x^4+18x6\) is irreducible in
\(\mathbb{Z}[i][x]\). Defined an \(A\)module, and proved that a function \(\cdot : A\times M\to M\) gives us an \(A\)module structure on \(M\) precisely when \(a\mapsto \ell_a\)
where \(\ell_a:M\to M, \quad \ell_a(m):=a\cdot m\), is a ring homomorphism from \(A\) to \({\rm End}(M)\). Pointed out that if \(F\) is a field, then an \(F\)module is
precisely an \(F\)vector space.
 2/1: We showed that there exists a bijection between ring homomorphism from \(A\) to \({\rm End}(M)\) and \(A\)module structures on \(M\). Defined the annahilator of
a module.
Pointed out that \(M\) can be viewed as an \(A/{\rm Ann}(M)\)module via the scalar multiplication given by \((a+{\rm Ann}(M))\cdot x:=a\cdot x\). Defined an \(A\)module
homomorphism and an \(A\)submodule. Discussed that a subset \(N\) of an \(A\)module \(M\) is a submodule if and only if it is a subgroup and closed under scalar multoplication.
Mentioned that \(N\) is an \(A\)submodule of \(M\) precisely when it is an \(A/{\rm Ann}(M))\)submodule of \(M\). Defined quotient of a module by one of its submodules.
Proved the first isomorphism theorem for modules. Defined direct product and external direct sum of a family of modules. Pointed out that they are the same precisely when the index
set is finite. Mentioned that \(\prod_{i=1}^\infty \mathbb{Z}/2\mathbb{Z}\) is in bijection with the power set of the set of positive integers, and so it is uncoutable, but
\(\bigoplus_{i=1}^\infty \mathbb{Z}/2\mathbb{Z}\) is in bijection with the set of positive squarefree integers, and so it is countable.
 2/3: Universal property of the external direct sum. Submodule generated by a subset. Sum of submodules. Internal direct sum. Free module generated by a set.
 2/6: Free module. \(A\)linear independence. The universal property of free modules. \(\mathfrak{a}M\) was defined. We discussed that \(f(\mathfrak{a}M)=\mathfrak{a}f(M)\) if
\(f:M\to N\) is an \(A\)module homomorphism. Mentioned that \({\rm ann}(M/\mathfrak{a}M)\supseteq \mathfrak{a}\), deduced that \(M/\mathfrak{a}M\) can be viewed as an
\(A/\mathfrak{a}\)module, and \(\overline{f}:M/\mathfrak{a}M\to f(M)/\mathfrak{a}f(M),\quad \overline{f}(x+\mathfrak{a}M):=f(x)+\mathfrak{a}f(M)\) is a surjective welldefined
\(A/\mathfrak{a}\)module homomorphism. For an ideal \(\mathfrak{a}\) of \(A\), \(\mathfrak{a} A^n=\mathfrak{a}^n\) and \(A^n/\mathfrak{a}A^n\simeq (A/\mathfrak{a})^n\).
 2/8: Proved that \(A^n\simeq A^m\) implies that \(n=m\). Defined rank of a module. Proved that \({\rm rank}_A(A^n)=n\) for every integral domain \(A\). Formulated the result
on the structure of submodules of \(D^n\) where \(D\) is a PID. Mentioned that the general idea behind the proof is related to Minkowski's reduction theory.
Went through some of the steps of the proof: (1) for every \(\phi\in {\rm Hom}_D(D^n,D), \phi(M)=\langle a_\phi\rangle\) and since \(D\) is Noetherian,
\(\Sigma:=\{\langle a_{\phi}\rangle\mid \phi\in {\rm Hom}_D(D^n,D)\}\) has a maximal element \(\langle a_{\phi_1}\rangle\). (2) Suppose \(a_{\phi_1}=\phi_1(y_1)\) for some \(y_1\in M\).
Then for every \(\phi\in {\rm Hom}_D(D^n,D)\), \(a_{\phi_1}\phi(y_1)\). (3) Used the previous step and showed that \(y_1=a_{\phi_1} x_1\) for some \(x_1\in D^n\) and \(\phi_1(x_1)=1\).
To see the parallel with Minkowski's reduction theory and the projection to rational directions, notice that for every \(\phi\in {\rm Hom}_D(D^n,D)\),
\(\phi(d_1,\ldots,d_n)=c_1d_1+\cdots+c_nd_n\) where \(c_i:=\phi(e_i)\). So \(\phi(M)\) is a scaled version of orthogonal projection to the line passing through \(\mathbf{v}:=(c_1,\ldots,c_n)\).
This point of view can help with the next step which is intersection with the space orthogonal to \(\mathbf{v}\).
 2/10: Proved that for every submodule \(M\) of \(D^n\) where \(D\) is a PID, there exist \(x_1,\ldots,x_n\in D^n\) and \(a_1,\ldots,a_m\in D\setminus\{0\}\) such that

\(a_1\cdotsa_m\).

\(D^n=Dx_1\oplus \cdots \oplus Dx_n\).

\(M=Da_1x_1\oplus \cdots \oplus Da_mx_m\).
In particular, every submodule of \(D^n\) is a free \(D\)module. Used this to show that every finitely generated module \(M\) over a PID \(D\) is isomorphic to
\(
D^r\oplus D/\langle a_1\rangle \oplus \cdots \oplus D/\langle a_m\rangle
\)
for some nonnegative integer \(r\) and \(a_1\cdotsa_m\) in \(D\setminus \{0\}\). \(a_i\)'s are called the invariant factors of \(M\). We used this to show that every
finitely generated module \(M\) over a PID \(D\) is isomorphic to \(D^r\oplus {\rm Tor}(M)\). In particular, \(M\) is torsionfree if and only if it is free.
We have also pointed out that \({\rm ann}({\rm Tor}(M))=\langle a_m\rangle\) and \(r={\rm rank}(M/{\rm Tor}(M))\); and so \(r\) and \(\langle a_m\rangle\) are
uniquely determined by \(M\). These are part of the uniqueness part of the fundamental theorem of f.g. modules over a PID.

2/13: We proved the Chinese Remainder Theorem for PIDs. Used it to show the elementary divisor decomposition of a finitely generated module over a PID. For an irreducible
element \(p\) of \(D\), defined the \(p\)primary submodule \(M_p\) of \(M\), and showed that \(M=\bigoplus_p M_p\) is unique. Using the elementary divisor decomposition,
we showed that \(M_p\simeq D/\langle p^{k_1}\rangle \oplus \cdots \oplus D/\langle p^{k_m}\rangle\) for a nondecreasing sequence of positive integers \(k_i\).

2/15: midterm.

2/17: Finished uniqueness part of the fundamental theorems of finitely generated modules over PIDs (using \(\dim_{k(p)} p^{\ell}M/p^{\ell+1}M\)). Recall that if \(\mathfrak{B}\) is an \(F\)basis of \(V\), then
\([\cdot]_{\mathfrak{B}}:V\to F^d\) is an \(F\)linear isomorphism. If \(T:V\to V\) is an \(F\)linear map, then \([T]_{\mathfrak{B}}\in {\rm M}_n(F)\) is a matrix such
that \([T(v)]_{\mathfrak{B}}=[T]_{\mathfrak{B}}[v]_{\mathfrak{B}}\). We showed that \([T]_{\mathfrak{B}}\) and \([T]_{\mathfrak{B}'}\) are similar to each other if
\(\mathfrak{B}\) and \(\mathfrak{B}'\) are two \(F\)basis of \(V\). For \(A\in {\rm M}_n(F)\), we defined the \(F[x]\)module \(V_A\). Showed that \(V_A\simeq V_B\) as
\(F[x]\)module if and only if \(A\) and \(B\) are similar in \({\rm M}_n(F)\). So to understand similarity of matrices it is enough to understand the \(F[x]\)module
structure of \(V_A\). Discussed why \(V_A\) is a torsion \(F[x]\)module. Deduce that there are unique monic polynomials \(f_1,\ldots,f_m\in F[x]\) such that
\(f_1\cdots f_m\) and
\(V_A\simeq F[x]/\langle f_1\rangle\oplus \cdots \oplus F[x]/\langle f_m\rangle\) as \(F[x]\)modules. Recalled that \(\mathfrak{B}:=\{1,x,\ldots,x^{d1}\}\) is an \(F\)basis of
\(F[x]/\langle f\rangle\) where \(d=\deg f\). Showed that the matrix representation of multiplication by \(x\) in \(F[x]/\langle f\rangle\) with respect to \(\mathfrak{B}\) is given
by the companion matrix of the monic polynomial \(f\).
 2/22: Finished the proof of the rational canonical form (existence and uniqueness). Defined the minimal polynomial of a matrix and showed that it is the same as
the largest invariant factor. Proved the characteristic polynomial of the companion matrix of a monic polynomial \(f\) is \(f\).
 2/24: Proved that the characteristic polynomial of a matrix is the product of its inavriant factors. Deduced the CayleyHamilton theorem, and the fact that every irreducible
factor of the characteristic polynomial is an irreducible factor of the minimal polynomial. Used elementary divisors and proved the existence and the uniqueness of the Jordan form
of a matrix. Used the uniqueness of Jordan from to show that a matrix is diagonalizable if and only if its minimal polynomial does not have a multiple zero. Defined exact sequence of
modules, short exact sequences (SES), and homomorphisms of SESs.
 3/1: Defined functors, opposite categories, mentioned examples of \({\rm SL}_n:{\rm Rng}\to {\rm Grp}\) and \({\rm GL_1}:{\rm Rng}\to {\rm Grp}\). Mentioned forgetful functors, and
defined representable functors. Defined natural transformations, mentioned \(\det: {\rm GL}_n\to {\rm GL}_1\) or \(\iota:{\rm SL}_n\to {\rm GL}_n\) as two examples. Mentioned Yoneda's lemma
(it will be part of your next HW). Proved that a representable functor \(h_M\) in the category of \(A\)modules can be viewed as a functor from the category of \(A\)modules to itself.
Proved that \(h_M\) is a left exact functor. Mentioned an example where \(h_M\) is not exact.
 3/3: Proved that the following are equivalent:
 \(h_M\) is an exact functor.
 \(h_M(f)\) is surjective if \(f:N\rightarrow N'\) is surjective.
 If \(f:N\to N'\) is surjective, then every \(\phi':M\rightarrow N'\) has a lift to \(\phi:M\to N\).
 Every SES of the form \(0\to N_1\to N_2\to M\to 0\) splits.
 \(M\) is a direct summand of a free module; that means \(M\oplus K\simeq F\) for some free module \(F\).
A module with these properties is called a projective module. Proved that if \(D\) is an integral domain, then free \(\Rightarrow\) projective \(\Rightarrow\) torsionfree.
If \(D\) is a PID, then free \(\Leftrightarrow\) projective \(\Leftrightarrow\) torsionfree. Proved that \(\langle 2,\sqrt{10}\rangle\) is not a free \(\mathbb{Z}[\sqrt{10}]\)module,
but it is a projective \(\mathbb{Z}[\sqrt{10}]\)module. Mentioned that this is true for every ideal, and it is mostly because this ring is locally a PID, and a finitely presented
module is projective if and only if it is locally free. (We did not prove these statements. In your HW assignment, you prove one direction.) Next we looked at the composite of two
representable functors \(h_{M_2}\circ h_{M_1}\). We want to show this is naturally isomorphic to a representable functor. We start by showing that there is a natural bijection between
\({\rm Hom}_A(M_1,{\rm Hom}_A(M_2,N))\) and \(A\)bilinear maps from \(M_1\times M_2\) to \(N\).
 3/6: Defined \(M_1\otimes_A M_2\). Proved that there exists a natural bijection between \(A\)bilinear maps \(f:M_1\times M_2\to N\) and \({\rm Hom}_A(M_1\otimes_A M_2,N)\).
For every \(A\)bilinear function \(f:M_1\times M_2\to N\), there exists a unique \(A\)module homomorphism \(\widehat{f}:M_1\otimes_A M_2\to N\) such that \(\widehat{f}(m_1\otimes m_2)=f(m_1,m_2)\)
(Universal property of tensor products). There exists a natural isomorphism between the functors \(h_{M_1}\circ h_{M_2}\) and \(h_{M_1\otimes_A M_2}\); alternatively there exists a
natural isomorphism \({\rm Hom}_A(M_1,{\rm Hom}_A(M_2,N))\simeq {\rm Hom}_A(M_1\otimes_A M_2, N)\). Mentioned that the tensor product of two free \(A\)modules is a free \(A\)module and
\({\rm rank}_A(F_1\otimes_A F_2)={\rm rank}_A(F_1){\rm rank}_A(F_2)\) if \(F_i\)'s are free \(A\)modules. Proved that \(\mathbb{Q}/\mathbb{Z}\otimes_{\mathbb{Z}} \mathbb{Q}=0\). Showed that
\(M\otimes_A A/\mathfrak{a}\simeq M/\mathfrak{a}M\).
 3/8: Proved that \(\mathbb{Z}/m\mathbb{Z}\otimes_\mathbb{Z} \mathbb{Z}/n\mathbb{Z}\simeq \mathbb{Z}/\gcd(m,n)\mathbb{Z}\). Defined \(f\otimes g\).
Defined the tensor functor \(T_M\). Proved that \(T_M\) is a right exact functor. Defined a flat module. Proved for an integeral domain \(D\) a flat \(D\)module is torsionfree. Then
proved that every free module is flat. In particular, if \(D\) is a PID and \(M\)is a finitely generated \(D\)module, then the following properties are equivalent:
(1) \(M\) is free, (2) \(M\) is projective, (3) \(M\) is flat, (4) \(M\) is torsionfree. Next we showed that \(M\oplus M'\) is flat if and only if \(M\) and \(M'\) are flat. This is used
to show that every projective module is flat.
 3/10: Using the fact that \(T_M\circ T_{M'}\) is naturally isomorphic to the functor \(T_{M\otimes_A M'}\), we deduce that the tensor product of two flat \(A\)modules is flat.
We mentioned that the converse is not true in general. But if \(D\) is a PID, \(M\) and \(M'\) are finitely generated \(D\)modules, and \(M\otimes_A M'\) is a nonzero flat module,
then \(M\) and \(M'\) are flat. We also mentioned that
\(\mathbb{Q}\otimes (\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z})\simeq
(\mathbb{Q}\otimes_{\mathbb{Z}} \mathbb{Z}) \oplus (\mathbb{Q}\otimes_{\mathbb{Z}} \mathbb{Z}/2\mathbb{Z}) \simeq \mathbb{Q}\) is flat, but \(\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}\)
is not flat. Next we defined the base change: if \(B\) is an \(A\)algebra, then \(T_B\) is a functor from \(A\)mod to \(B\)mod. Then tensor product of two \(A\)algebras was defined.
We stated the following isomorphism: if \(I\) is an ideal of \(A[x]\), then \(A[x]/I\otimes_A B\simeq B[x]/B\overline{I}\). We used this to prove:
 \(A[x]\otimes_A A[x]\simeq A[x,y]\).
 \(A[x]\otimes_A B\simeq B[x]\).
 \(\mathbb{Q}[i]\otimes_{\mathbb{Q}}\mathbb{Q}[i]\simeq \mathbb{Q}[i]\oplus \mathbb{Q}[i]\).
 \(\mathbb{Z}[i]\otimes_{\mathbb{Z}}\mathbb{F}_p\simeq \mathbb{F}_{p^2}\) if \(p\equiv 3\pmod{4}\), \(\simeq \mathbb{F}_p\oplus \mathbb{F}_p\) if \(p\equiv 1\pmod{4}\),
and \(\simeq \mathbb{F}_2[y]/\langle y^2\rangle\) if \(p=2\).
 3/13: An outline of the proof of \(A[x]/I\otimes_A B\simeq B[x]/B\overline{I}\) was mentioned. A field extension, subfield, and an extension of a field were defined. Algebraic
and transcendental elements were defined. We proved that \(\alpha\in E\) is algebraic over \(F\) if and only if \(1,\alpha,\ldots\) are \(F\)linearly dependent. Hence, if \(E/F\) is a finite
field extension, then every element of \(E\) is algebraic over \(F\). Considered the evaluation map \(e_{\alpha}:F[x]\to E\) and used it to show that there exists a monic irreducible polynomial
\(m_{\alpha,F}(x)\in F[x]\) such that for \(f(x)\in F[x]\), \(f(\alpha)=0\) if and only if \(m_{\alpha,F}(x)f(x)\); this monic polynomial is called the minimal polynomial of \(\alpha\) over \(F\).
Discussed that the image of \(e_{\alpha}\) is the subring of \(E\) which is generated by \(F\) and \(\alpha\) and it is denoted by \(F[\alpha]\). Proved that
\(F[\alpha]\simeq F[x]/\langle m_{\alpha,F}\rangle\) is a field. Showed that if \(\deg m_{\alpha,F}=d\), then \(\{1,\alpha,\ldots,\alpha^{d1}\}\) is an \(F\)basis of \(F[\alpha]\); in
particular, \(\dim_F F[\alpha]=\deg m_{\alpha,F}\).
 3/15: Proved the existence of a splitting field of \(f(x)\in F[x]\) over \(F\). Proved that if \(p(x)\) is irreducible in \(F[x]\), \(\theta:F\to F'\) is a field isomorphism,
\(E/F\) and \(E'/F'\) are field extensions, and \(\alpha\in E,\alpha'\in E'\) such that \(p(\alpha)=0\) and \(p^\theta(\alpha')=0\), then there exists a field isomorphism
\(\widehat{\theta}:F[\alpha]\to F'[\alpha']\) which is an extension of \(\theta\) and \(\widehat{\theta}(\alpha)=\alpha'\).
 3/17: Proved that if \(\theta:F\to F'\) is a field isomorphism, \(E\) is a splitting field of \(f\) over \(F\), and \(E'\) is a splitting field of \(f^\theta\) over \(F'\), then
there exists a field isomorphism \(\widehat{\theta}:E\to E'\) which is an extension of \(\theta\); moreover if \(\alpha\in E\) is a zero of an irreducible factor \(p\) of \(f\)
and \(\alpha'\in E'\) is a zero of \(p^\theta\), then there exists an extension \(\widehat{\theta}:E\to E'\) of \(\theta\) such that \(\widehat{\theta}(\alpha)=\alpha'\). Next
we investigated finite fields. We proved that if \(F\) is a finite field, then as an additive group it is a vector space over \(\mathbb{F}_p\) for a prime \(p\); in particular,
\(F=p^d\) for some positive integer \(d\). Then we went over the proof of why the group of units of \(F\) is cyclic. Next we proved that
\(x^{p^d}x=\prod_{\alpha\in F}(x\alpha)\) in \(F[x]\) if \(F=p^d\). Then we proved that for every prime \(p\) and positive integer \(d\), there exists a unique,
up to an isomorphism, finite field of order \(p^d\). We denote this field by \(\mathbb{F}_{p^d}\). We proved that \(\mathbb{F}_{p^d}\) is a splitting field of \(x^{p^d}x\)
over \(\mathbb{F}_p\). We went over derivative of a polynomial in \(A[x]\), and showed that if \(D\) is an integeral domain, then for \(\alpha\in D\) and \(f\in D[x]\),
\(f(x)\) is a multiple of \((x\alpha)^2\) if and only if \(f(\alpha)=f'(\alpha)=0\). After this, we proved that the tower formula: \([K:F]=[K:E][E:F]\) if
\(F\subseteq E\subseteq K\) is a tower of fields; in particular, \(K/F\) is a finite extension if and only if \(K/E\) and \(E/F\) are finite extensions. We ended our lecture by
showing that \(\mathbb{Q}[\sqrt[p]{2},\zeta_p]\) is a splitting field of \(x^p2\) over \(\mathbb{Q}\) and \([\mathbb{Q}[\sqrt[p]{2},\zeta_p]:\mathbb{Q}]=p(p1)\) if \(p\) is prime.

Exams.
 Midterm:
 Time: Feb 15, 2023.
 Location: this is an inclass exam.
 Topics: All the topics that are discussed in class, till the end of Lecture on Feb 10.
 Questions are fairly similar to the homework assignments
and the examples discussed in the class. Make sure that you know how to solve anyone of them.
 Here are the grades: 37, 31.5, 28.5, 28, 28, 27, 27, 26.5, 26.5, 24, 23.5, 23, 22.5, 21.5,
20.5, 20, 18.5, 18, 17, 16.5, 16.5, 15.5, 14, 13, 12.5, 12, 6.25.
 Here are the medians of each problem: (1) 10, (2) 7, (3) 5, (4) 2.
 I can see two reasons for the lower than expected median for problem 4:
(1) time pressure (2) we did not have a HW assignment for that topic;
and so you did not have enough experience with that type of problem.

I consider a score above 26 as a very good performance in this exam (PhD), a score above 19.5 as good (PhD),
and above 15 OK (Ms). If your score is below 15, please send me an email and we can chat a bit.
 The final exam:
 Date: 3/22/2023, 811am.
 Location: In class.
 Topics: All the topics that were discussed in class, your homework assignments,
and relevant examples and exercises in your book.
 Questions are fairly similar to the
homework assignments and the examples discussed in the class. Make sure
that you know how to solve anyone of them.
 Here are the grades (out of 90): 90, 89, 89, 87, 87, 86, 82.5, 82, 80, 77, 76,
65, 61.5, 60, 57, 57,
50.5, 50, 49, 45, 45, 43.5, 43.5, 42, 39, 37, 15.5.
 Here are the median and mean of each problem: (1) 15, 13.54, (2) 8, 7.04, (3) 8, 8.04, (4) 6, 5.21,
(5) 7.5, 6.25, (6) 8, 9.29, (7) 1, 3.91, (8) 9.75, 7.63.

Students who scored above 75, with a similar performance in the qual, would get a PhD pass.
Students who scored from 57 to 65, with a similar performance in the qual, would get a Provisional PhD pass.
Students who scored from 45 to 51, with a similar performance in the qual, would get a Master pass.
Students who scored from 42 to 45, it is not clear; you might or might not pass.
No grad student received below 40, but a similar performance in the qual would have resulted in failing the exam.
