|
My notes.
You can use my lecture notes from 2018 and
2019.
I will summarize what gets covered during the lectures here. You should go over the notes and your book on the relevant topics.
-
1/5 (1) Recalled \(v_p\), \({\rm sgn}\), \(|\cdot |\), \(\gcd \), and their basic properties.
Defined the content of a polynomial and primitive polynomial. Proved that product of two primitive polynomials is primitive (Gauss's lemma version 1). Proved that
\(\alpha(fg)=\alpha(f)\alpha(g)\) (Gauss's lemma version 2).
Almost proved that if \(f\in D[x]\setminus D\), \(g_1,g_2\in F[x]\) where \(F\) is the field of fractions of \(D\), and \(f=g_1g_2\), then there exist
\(c_1,c_2\in F^\times\) such that \(c_1c_2=1\) and \(c_1g_1, c_2g_2\in D[x]\) (Gauss's lemma version 3).
-
1/7 (2) Finished proof of Gauss's lemma version 3. Proved that for \(f,g\in D[x]\), \(f|g\) in \(F[x]\) if and only if \(f|g\) in \(D[x]\). Showed that for \(a\in D\),
\(a\) is prime in \(D\) if and only if \(a\) is prime in \(D[x]\). Next, proved that \(a\) is irreducible in \(D\) if and only if \(a\) is irreducible in \(D[x]\).
Proved that if \(f\) is primitive of positive degree, then \(f\) is irreducible in \(D[x]\) if and only it is irreducible in \(F[x]\). Almost proved that if \(D\) is a UFD, then
\(D[x]\) is a UFD.
-
1/9 (3) Completed proof of: \(D\) UFD implies that \(D[x]\) is a UFD. Proved mod \(p\) irreducibility criterion and Eisenstein's irreducibility criterion. Defined an
\(A\)-module. Mentioned some examples: vector spaces, ideals, direct product and external direct sum.
-
1/12 (4)
\({\rm Hom}_A(M,N)\) is an \(A\)-module if \(A\) is a commutative ring. There are natural bijections \({\rm Hom}_A(N,\prod_{i\in I}M_i)\simeq \prod_{i\in I} {\rm Hom}_A(N,M_i)\) and
\({\rm Hom}_A(\bigoplus_{i\in I}M_i,N)\simeq \prod_{i\in I} {\rm Hom}_A(M_i,N)\). Defined the quotient of a module by one of its submodules. Stated the first isomorphism theorem
for modules.
-
1/14 (5)
Proved the first isomorphism theorem. Mentioned that every vector space has a basis and argued why this is the case using Zorn's lemma. Used that as a motivation to define the rank
of a module and the minimum number of generators of a module. Pointed out that in your HW assignments you will prove \({\rm rank}(M)\leq d(M)\) for every module and they are equal to
\(n\) if and only if \(M\simeq A^n\). Mentioned that \({\rm rank}(A^n)=n\). Stated results that if there exists a surjective \(A\)-module homomorphism \(A^n\to A^m\), then \(n\geq m\).
And if there exists an injective \(A\)-module homomorphism \(A^n\to A^m\), then \(n\leq m\). The first one is easier than the second one, and you will prove both results as part of your HW assignments.
Next, we proved the existence of a free module generated by a set. Mentioned sum of a family of submodules. Defined internal direct sum of submodules, and proved its main properties.
-
1/16 (6)
Proved that \(m_1,\ldots,m_n\) are \(A\)-linearly independent if and only if \(A^n\to \langle m_1,\ldots,m_n\rangle, \quad (a_1,\ldots,a_n)\mapsto \sum_{i=1}^n a_im_i\) is an isomorphism.
Mentioned that every \(A\)-linear map \(\theta\colon A^n\to A^m\) is in one-to-one correspondence to an \(m\)-by-\(n\) matrix \([\theta]\) such that
\(|\theta(v)\rangle=[\theta] |v\rangle \).
Used this to show that if \(\theta\) is surjective, then \(n\geq m\).
Deduced that if \(A^n\simeq A^m\), then \(m=n\).
Proved that \({\rm rank}_D(D^n)=n\) if \(D\) is an integral domain.
Proved that every non-zero ideal of \(A\) is a free \(A\)-module of rank 1 if and only if \(A\) is a PID.
Stated the important theorem on classification of submodules of \(D^n\) assuming \(D\) is a PID.
Mentioned the general ideal of Minkowski in order to describe discrete subgroups of \(\mathbb{R}^n\): Minkowski's successive minima.
-
1/21 (7)
Proved that if \(D\) is a PID and \(M\) is a submodule of \(D^n\), then there exist \(x_1,\ldots,x_n\in D^n\) and \(a_1,\ldots,a_m\in D\setminus \{0\}\) such that
- \(D^n=Dx_1\oplus \cdots \oplus Dx_n\).
- \(a_1|\cdots |a_m\).
- \(M=Da_1x_1\oplus \cdots \oplus Da_mx_m\).
-
1/23 (8)
Proved that if \(D\) is a PID and \(M\) is a finitely generated \(D\)-module,
then there exist a non-negative integer \(r\) and
\(a_1,\ldots,a_m\in D\setminus (D^\times \cup \{0\})\)
such that
- \(M\simeq M(r;a_1,\ldots,a_m)\) where
\(M(r;a_1,\ldots,a_m)=D^r\oplus \frac{D}{\langle a_1\rangle} \oplus \cdots \oplus \frac{D}{\langle a_m\rangle}\).
- \(a_1|\cdots |a_m\).
This is the existence part of the fundamental theorem of finitely generated modules over a PID.
Next, we worked towards the uniqueness.
We proved that \({\rm rank}(M(r;a_1,\ldots,a_m))=r\).
Defined the torsion submodule \({\rm Tor}(M)\) of a \(D\)-module \(M\) where \(D\) is an integral domain.
Pointed out that if \(M\simeq M'\), then \({\rm Tor}(M)\simeq {\rm Tor}(M')\).
Proved that \({\rm Tor}( M(r;a_1,\ldots,a_m))=\{0\}^r\oplus M(0;a_1,\ldots,a_m) \).
Then, we proved the generalized Chinese Remainder Theorem: suppose \(A\) is a unital commutative ring,
\(\mathfrak{a}_i\)'s are pairwise coprime ideals meaning that for \(i\neq j\),
\(\mathfrak{a}_i+\mathfrak{a}_j=A\). Then
\(\frac{A}{\mathfrak{a}_1\cap \cdots \cap \mathfrak{a}_m}\simeq \frac{A}{\mathfrak{a}_1}\oplus \cdots \oplus \frac{A}{\mathfrak{a}_m}\) as \(A\)-algebras.
Pointed out that in a PID, \(\langle a\rangle +\langle b\rangle =\langle {\rm gcd}(a,b)\rangle\), and so the concept of being coprime is compatible with the classical definition of having \({\rm gcd}(a,b)=1\).
-
1/26 (9)
Proved that if \(D\) is a PID, then a finitely generated \(D\)-module is torsion-free if and only if it is free.
Argued that if \(D\) is an integral domain, then the rank of every non-zero ideal is \(1\) and every non-zero ideal is torsion-free. Hence, an ideal is free if and only if it is a principal ideal. In particular, if
\(D\) is an integral domain which is not a PID, then it has a torsion-free ideal which is not free.
Proved that \(b\cdot \frac{A}{\mathfrak{a}}\simeq \frac{\langle b\rangle +\mathfrak{a}}{\mathfrak{a}}\) as \(A\)-modules.
Pointed out that \(b\cdot \frac{D}{\langle a\rangle}=\frac{\langle {\rm gcd}(a,b)\rangle}{\langle a\rangle}\) if \(D\) is a PID.
Moreover, \(b\cdot \frac{D}{\langle a\rangle}\simeq \frac{D}{\langle a/{\rm gcd}(a,b)\rangle }\).
Using generalized CRT and the above argument, we proved that
\(
\dim_{k(p)} \frac{p^r\cdot M(0;a_1,\ldots,a_m)}{p^{r+1}\cdot M(0;a_1,\ldots,a_m)}=|\{i\mid v_p(a_i)>r\}|.
\)
Using this, argued why the Young diagram given by the numbers \(\dim_{k(p)} \frac{p^r\cdot M(0;a_1,\ldots,a_m)}{p^{r+1}\cdot M(0;a_1,\ldots,a_m)}\) is the
conjugate of the Young diagram associated to \(v_p(a_i)\)'s.
-
1/28 (10)
Mentioned that the module structure of an \(F[x]\)-module \(M\) is uniquely determined by its vector space structure and the \(F\)-linear map
\(\ell_x\colon M\to M, \quad \ell_x(m):=x\cdot m\). Hence, for two \(F[x]\)-modules \(M_1\) and \(M_2\), an \(F\)-linear map \(\theta\colon M_1\to M_2\) is
an \(F[x]\)-module homomorphism if and only if \(\ell_x\circ \theta=\theta\circ \ell_x\).
For a matrix \(a\in {\rm M}_n(F)\), let \(V_a\) be the \(F[x]\)-module such that \(V_a=F^n\) and \(\ell_x(v):=a|v\rangle\). Hence, for every polynomial \(f(x)\in F[x]\) and
\(v\in F^n\), \(f(x)\cdot v:=f(a)|v\rangle\). We proved that \(V_a\simeq V_b\) as \(F[x]\)-modules if and only if \(a\) is similar to
\(b\). We also showed that \(V_a\oplus V_b\simeq V_{{\rm diag}(a,b)}\). Using the classification of finitely generated modules over a PID, we proved that there exist unique
monic non-constant polynomials \(f_1,\ldots, f_m\) such that \(f_1|\cdots |f_m\) and \(V_a\simeq \frac{F[x]}{\langle f_1\rangle}\oplus \cdots \oplus \frac{F[x]}{\langle f_m\rangle}\)
as \(F[x]\) modules. These polynomials are called the invariant polynomials of \(a\). Hence, two matrices are similar if and only if they have the same set of invariant factors. Next, we proved that, for a monic non-constant polynomial \(f\), \(F[x]/\langle f\rangle\simeq V_{c_f}\), where
\(
c_f=
\begin{pmatrix}
0 & 0 & \cdots & 0 & -a_{0} \\
1 & 0 & \cdots & 0 & -a_{1} \\
0 & 1 & \cdots & 0 & -a_{2} \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & 1 & -a_{n-1}
\end{pmatrix}.
\)
Hence, \(V_a\simeq V_{f_1}\oplus \cdots \oplus V_{f_m}\) where \(f_i\)'s are the invariant factors of \(a\).
-
1/30 (11)
Proved the existence and uniqueness of the rational canonical form of a matrix.
Defined and proved the main property of the minimal polynomial of an element in a finite dimensional \(F\)-algebra.
Proved that the minimal polynomial of a matrix over a field is the same as the last invariant polynomial of that matrix.
Defined the characteristic polynomial of a matrix.
Pointed out that \(\chi_{gag^{-1}}=\chi_a\).
Proved that \(\chi_{c_f}=f\) for every monic non-constant polynomial \(f\).
Proved that the characteristic polynomial of a matrix over a field is the product of its invariant factors.
Deduced the Cayley-Hamilton theorem.
Proved that the minimal polynomial and the characteristic polynomial of a matrix over a field have the same set of irreducible factors.
-
2/2 (12)
Proved the existence and uniqueness of Jordan forms.
Proved that a matrix is diagonalizable over a field \(F\) if and only if its minimal polynomial factors into degree one factors over \(F\) and all of its zeros are distinct.
Defined sequences, complexes, exact sequences, and short exact sequences of modules.
Defined homomorphisms of complexes and short exact sequences.
Proved that every SES is isomorphic to a standard SES.
Defined a standard SES associated with an external direct sum.
-
2/4 (13)
For a given SES \(0\to M_1\xrightarrow{f_1} M_2 \xrightarrow{f_2} M_3\to 0\) the following statments are equivalent:
- There exists a submodule \(N_2\) of \(M_2\) such that \(M_2=f_1(M_1)\oplus N_2\).
- There exists an \(A\)-module homomorphism \(g_1\colon M_2\to M_1\) such that \(g_1\circ f_1={\rm id}_{M_1}\).
- There exists a SES isomorphism \( ({\rm id}_{M_1},\theta,{\rm id}_{M_3})\) between the given SES and \(0\to M_1\to M_1\oplus M_3\to M_3\to 0\).
- There exists an \(A\)-module homomorphism \(g_2\colon: M_3\to M_2\) such that \(f_2\circ g_2={\rm id}_{M_3}\).
If the above conditions hold, we say this is a splitting SES.
Next, we defined categories, mentioned various examples: Set, Gp, Ab, \(A\)-mod, Ring. We also constructed a category using a given monoid.
After that, we defined morphism between categories, and called them functors. We also defined contravariant functors.
Finally mentioned the example of zeros of integer polynomials as a functor from the category of unital commutative rings to the category of sets.
-
2/6 (14)
Mentioned that \({\rm SL}_n\), \({\rm GL}_1\), and \({\rm GL}_n\) are functors from the category of unital commutative rings to the category of groups.
Defined representable functor and contravariant functor: \(h_x\colon \mathcal{C}\to \underline{\rm Set}\) and \(h^x\colon \mathcal{C}\to \underline{\rm Set}\).
Enriched representable functor and contravaiant functor of \(A\)-modules to obtain functors from \(A\)-mod to itself.
Think about representable functors as a way of using observers to understand objects.
Proved that for given arrows \(N_1\xrightarrow{f_1}N_2\xrightarrow{f_2} N_3\) in the category of \(A\)-modules:
- If for all \(A\)-modules \(M\), \(h_{M}(N_1)\xrightarrow{h_M(f_1)}h_M(N_2)\xrightarrow{h_M(f_2)}h_M(N_3)\) is exact, then \(N_1\xrightarrow{f_1}N_2\xrightarrow{f_2} N_3\) is exact.
- If for all \(A\)-modules \(M\), \(h^{M}(N_1)\xleftarrow{h^M(f_1)}h^M(N_2)\xleftarrow{h^M(f_2)}h^M(N_3)\) is exact, then \(N_1\xrightarrow{f_1}N_2\xrightarrow{f_2} N_3\) is exact.
Defined the phrases: a homomorphism has a lift and a homomorphism factors through another one . For \(N_1\xrightarrow{f} N_2\), \(g\colon M\to N_2\) is in the image of \(h_M(f)\) if and only if
it has a lift \(\widetilde{g}\colon M\to N_1\) along \(f\), and \(h\colon N_1\to M\) is in the image of \(h^M(f)\) if and only if it factors through \(f\).
To investigate the converse direction, we proved that
- If \(f\) is injective, then \(h_M(f)\) is injective.
- If \(f\) is surjective, then \(h^M(f)\) is injective.
Here you can find proof of the last statement: proving exactness using observers.
-
2/9 (15)
Proved that if \(D\) is an integral domain, \(T\) is a torsion \(D\)-module and \(M\) is a torsion-free \(D\)-module, then \(h_T(M)=0\). Deduce that \(h_{\mathbb{Z}/n\mathbb{Z}}(\pi_n)\) is not surjective
where \(\pi_n\colon \mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}\) is the surjective quotient map.
Defined a divisible abelian group. Mentioned that \(\mathbb{Q}\) is a divisible group. Proved that if \(J\) is a divisible group, \(T\) is an abelian group and \(nT=0\), then \(h^T(J)=0\). Deduced that
\(h^{\mathbb{Z}/n\mathbb{Z}}(i)\) is not surjective where \(i\colon \mathbb{Z}\to \mathbb{Q}\) is the inclusion map.
Proved that \(h_M\) and \(h^M\) are left-exact.
Defined an exact additive functor.
Proved that the following statements are equivalent:
- \(h_M\) is an exact functor.
- \(h_M(f)\) is surjective if \(f\) is surjective.
- If \(f\colon N_1\to N_2\) is surjective, then every \(g\colon M\to N_2\) has a lift along \(f\).
- Every SES of the form \(0\to N_1\to N_2\to M\to 0\) splits.
- There exists a free \(A\)-module \(F\) and an \(A\)-module \(K\) such that \(F\simeq M\oplus K\). (We say \(M\) is a direct summand of a free module.)
We say \(M\) is a projective \(A\)-module if the above statements hold.
-
2/11
Midterm.
-
2/13 (16)
Finshed the proof of properties of a projective module.
Deduced that free implies projective.
Mentioned that over an integral domain, projective implies torsion-free.
Over a PID, a f.g. torsion-free module is free. Hence, for a f.g. module over a PID, these concepts are equivalent: torsion-free, projective, and free.
Proved that if \(\mathfrak{a}\) is a non-zero finitely generated ideal of an integral domain \(D\), then \(\mathfrak{a}\) is projective if and only if there exists
a finitely generated \(D\)-submodule \(M\) of a field of fraction of \(D\) such that \(\mathfrak{a}M=D\).
Used this to show that \(\langle x,2\rangle \) is not a projective \(\mathbb{Z}[x]\)-module. We mentioned that \(\langle 2,\sqrt{-10}\rangle\) is a projective \(\mathbb{Z}[\sqrt{-10}]\)-module which is not free.
-
2/18 (17)
Proved that \(\mathbb{Q}\) is torsion-free but not projective.
Defined natural transformation. Mentioned that a natural transformation should be viewed as a homomorphism between functors.
Mentined that \(\det\colon {\rm GL}_n\to {\rm GL}_1\) and \(\iota\colon {\rm SL}_n\to {\rm GL}_n\) are natural transformations.
Defined (and proved) the functor \(B_{M_1,M_2}\) of bilinear \(A\)-linear maps from the category of \(A\)-modules to the category of \(A\)-modules.
Proved that \(h_{M_2}\circ h_{M_1}\simeq B_{M_1,M_2}\). In particular, \(h_{M_1}\circ h_{M_2}\simeq h_{M_2}\circ h_{M_1}\).
-
2/20 (18)
Using Yoneda's lemma, we motivated definition of the tensor product of two modules.
Defined tensor product of two modules. Mentioned that it is generated by its pure tensor elements.
The map \(\iota\colon M_1\times M_2\to M_1\otimes_A M_2,\quad \iota(m_1,m_2):=m_1\otimes m_2\) is \(A\)-bilinear. It is \(A\)-balanced.
Proved the universal property of tensor products. Mentioned that it is the same as saying \(B_{M_1,M_2}\simeq h_{M_1\otimes_A M_2}\).
Proved that \(h_{M_2}\circ h_{M_1}\simeq h_{M_1\otimes_A M_2}\).
Mentioned that \(h_x\simeq h_y\) implies \(x\simeq y\). Deduced that \(M_1\otimes_A M_2\simeq M_2\otimes_A M_1\).
Proved that the tensor product of two projective modules is projective.
Proved that if \(D\) is an integral domain, \(M\) is a torsion \(D\)-module and \(F\) is a field of fractions of \(D\), then \(F\otimes_D M=0\).
Stated that \(T_M(N):=M\otimes_A N\) defines a functor and \({\rm Hom}_A(T_M(N),N')\simeq {\rm Hom}_A(N,h_M(N'))\). Defined left and right adjoint functors.
Stated that \(A/\mathfrak{a}\otimes_A M\simeq M/\mathfrak{a} M\) and used it to prove \(\mathbb{Z}/n\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Z}/m\mathbb{Z}\simeq \mathbb{Z}/\gcd(m,n) \mathbb{Z}\).
-
2/23 (19)
Proved that \(A/\mathfrak{a}\otimes_A M\simeq M/\mathfrak{a} M\) and the isomorphism is given by \((x+\mathfrak{a})\otimes m\mapsto xm+\mathfrak{a}M\).
Defined the tensor product of two \(A\)-module homomorphisms.
Proved that \((f\otimes g)\circ (f'\otimes g')=(f\circ f') \otimes (g\circ g')\) whenever the composites exist.
Defined the tensor functor \(T_M\).
Mentioned that \(M\mapsto M/\mathfrak{a}M\) can be made into a functor \(F_{\mathfrak{a}}\), and \(T_{A/\mathfrak{a}}\simeq F_{\mathfrak{a}}\). In particular, \(T_A\) is isomorphic to the identity functor.
Pointed out that \(h^N\circ T_M\simeq h^{h_M(N)}\).
Used this to show \(h^N\circ (T_{M_1}\circ T_{M_2})\simeq h^N \circ T_{M_1\otimes_A M_2}\) for all \(N\). Deduced that \(T_{M_1}\circ T_{M_2}\simeq T_{M_1\otimes_A M_2}\).
In particular, \(M_1\otimes_A(M_2\otimes_A M_3)\simeq (M_1\otimes_A M_2)\otimes_A M_3\).
Noticed that \(\prod_I h_{M_i}\simeq h_{\bigoplus_I M_i}\) and \(\prod_I h^{M_i}\simeq h^{\prod_I M_i}\).
Used this to show that \(M\otimes_A (\oplus_I M_i)\simeq \bigoplus_I M\otimes_A M_i\).
-
2/25 (20)
Proved that \(T_{\oplus_{i\in I} M_i}\simeq \bigoplus_{i\in I} T_{M_i}\).
Proved that if \(F\) and \(G\) are additive functors and \(F\) is a left-adjoint of \(G\), then \(F\) is right-exact.
Deduced that \(T_M\) is right-exact. Defined a flat module.
Proved that \(\bigoplus_{i\in I} M_i\) is flat if and only if \(M_i\) is flat for all \(i\).
Proved that a projetive module is flat.
Mentioned that \(T_{S^{-1}A}\simeq S^{-1}\underline{.}\) and deduced that \(S^{-1}A\) is a flat \(A\)-module.
Obtained that \(\mathbb{Q}\) is a flat \(\mathbb{Z}\)-module but it is not projective.
Proved that if \(D\) is an integral domain and \(M\) is a flat \(D\)-module, then \(M\) is torsion-free.
Deduced that \(\mathbb{Z}/n\mathbb{Z}\) is not a flat \(\mathbb{Z}\)-module.
Mentioned that for a finitely presented module, the following are equivalent: projective, locally free, and flat.
Used \(T_{M_1}\circ T_{M_2}\simeq T_{M_1\otimes_A M_2}\) to deduce that the tensor product of two flat modules is flat.
Proved that if \(B\) is an \(A\)-algebra, then \(T_B\) can be viewed as a functor from the category of \(A\)-modules to the category of \(B\)-modules.
-
2/27 (21)
Mentioned the forgetful functor from the category of \(B\)-modules to the category of \(A\)-modules assuming that \(B\) is an \(A\)-algebra.
Stated the result that \(B\otimes_A C\) is an \(A\)-algebra if \(B\) and \(C\) are \(A\)-algebras with \((b\otimes c)(b'\otimes c')=(bb')\otimes (cc')\).
Stated the theorem: \(A[x]/\mathfrak{a}\otimes_A B\simeq B[x]/\mathfrak{a}^{\rm e}\) as \(B\)-algebras.
Used this result to prove \(\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}\simeq \mathbb{C}\oplus \mathbb{C}\) as \(\mathbb{C}\)-algebras. We showed that an isomorphism is given by
\(z\otimes z'\mapsto (z'z,z'\overline{z})\).
Proved that \(\mathbb{Z}[i]\otimes_{\mathbb{Z}} \mathbb{Z}/p\mathbb{Z}\simeq \mathbb{Z}/p\mathbb{Z}\oplus \mathbb{Z}/p\mathbb{Z}\) if \(p\) is a prime which is \(1\) modulo \(4\), this tensor product is
a field of order \(p^2\) is \(p\) is a prime which is \(3\) modulo \(4\), and it has a nilpotent element if \(p=2\).
We went half way through the proof of the mentioned isomorphism of \(B\)-algebras.
-
3/2 (22)
Completed the proof of \(A[x]/\mathfrak{a}\otimes_A B\simeq B[x]/\mathfrak{a}^{\rm e}\) as \(B\)-algebras.
Defined field extensions, \(\theta\)-embedding of \(E\) to \(E'\) for given field extensions \(E/F\) and \(E'/F'\) and an isomorphism \(\theta\colon F\to F'\).
Defined algebraic and transcendental elements.
Proved that if \(E/F\) is a field extension and \(\alpha\in E\) is algebraic over \(F\), then there exists a unique monic non-constant polynomial \(m_{\alpha;F}(x)\in F[x]\) such that
for every \(f\in F[x]\), \(f(\alpha)=0\) if and only if \(m_{\alpha;F}(x)|f(x)\) in \(F[x]\). This polynomial is called the minimal polynomial of \(\alpha\) over \(F\).
Proved that \(m_{\alpha;F}(x)\) is irreducible in \(F[x]\). We also proved that if \(p(x)\in F[x]\) is irreducible, monic, and \(p(\alpha)=0\), then \(p(x)=m_{\alpha;F}(x)\).
Next, we proved that if \(\alpha\in E\) is algebraic over \(F\), then
- The subring \(F[\alpha]\) generated by \(F\) and \(\alpha\) is a field.
- \((1,\alpha,\ldots,\alpha^{d-1})\) is an \(F\)-basis of \(F[\alpha]\) where \(d=\deg m_{\alpha;F}\). In particular, \(\dim_F F[\alpha]=\deg(m_{\alpha;F})\).
These results are proved using the evaluation at \(\alpha\) map and the isomorphism \(F[x]/\langle m_{\alpha;F}(x)\rangle\simeq F[\alpha]\).
We defined the degree of a field extension \(E/F\) as \(\dim_F E\) and it is denoted by \([E:F]\).
Defined finite and algebraic extensions.
Next, we formulated the following question which will be addressed in the next lecture: given a non-constant polynomial \(f\in F[x]\) can we find a pair \((E/F,\alpha)\) such that
- \(E/F\) is a field extension,
- \(\alpha\in E\) and \(f(\alpha)=0\), and
- \(E=F[\alpha]\).
We noted that an affirmative answer in the case where \(f\) is irreducible implies the general case.
-
3/4 (23)
Proved the tower formula: if \(E/F\) and \(K/E\) are two field extensions, then \([K:F]=[K:E][E:F]\). In particular, \(K/F\) is a finite extension if and only if both \(K/E\)
and \(E/F\) are finite extensions.
Proved that if \(f\in F[x]\) is irreducible, then there exists \((E/F,\alpha)\) such that
- \(E/F\) is a field extension, \(\alpha\in E\), and \(E=F[\alpha]\).
- \(f(\alpha)=0\).
Proved that the following holds: suppose \(\theta\colon F\to F'\) is a field isomorphism, \(f\in F[x]\) is irreducible, \((E/F,\alpha)\) is assigned to \(f\) and \((E'/F',\alpha')\) is assigned to
\(f^\theta\) as in the previous statement. Then there exists a unique \(\theta\)-isomorphism \(\widehat{\theta}\colon E\to E'\) such that \(\widehat{\theta}(\alpha)=\alpha'\).
Proved the existence of a splitting field: for a polynomial \(f\in F[x]\) of degree \(d\), there exists a pair \((E/F,[\alpha_1,\ldots,\alpha_d])\) with the following properties:
- \(E/F\) is a field extension, \([\alpha_1,\ldots,\alpha_d]\) is a multiset, and \(E=F[\alpha_1,\ldots,\alpha_d]\).
- \(f(x)=c\prod_{i=1}^d(x-\alpha_i)\) where \(c\in F\) is the leading coefficient of \(f\).
We call \((E/F,[\alpha_1,\ldots,\alpha_d])\) a splitting data of \(f\) over \(F\), and \(E\) is called a splitting field of \(f\) over \(F\).
Next, we stated the following uniqueness of splitting data: suppose \(\theta\colon F\to F'\) is a field isomorphism and \(f\in F[x]\) is a polynomial of degree \(d\).
Let \((E/F,[\alpha_1,\ldots,\alpha_d])\) and \((E'/F',[\alpha_1',\ldots,\alpha_d'])\) be splitting datas of \(f\) and \(f^\theta\), respectively. Then there exists a
\(\theta\)-isomorphism \(\widehat{\theta}\colon E\to E'\), and for such an isomorphism, we have \(\widehat{\theta}([\alpha_1,\ldots,\alpha_d])=[\alpha_1',\ldots,\alpha_d']\).
-
3/6 (24)
We finished proof of the uniqueness of a splitting data of a polynomial.
For a polynomial \(f\in F[x]\), defined \(V_f(E):=\{\alpha\in E\mid f(\alpha)=0\}\). Proved that if \(E/F\) and \(E'/F'\) are field extensions,
\(\theta\colon F\to F'\) is a field isomorphism, and \(f\in F[x]\), then for every \(\theta\)-isomorphism \(\widehat{\theta}\), \(\widehat{\theta}(V_f(E))\subseteq V_{f^{\theta}}(E')\).
Proved that if \(F\) is a finite field, then its order is a power of a prime \(p\) and it is a field extension of a finite field \(\mathbb{F}_p\) of order \(p\).
Proved that the multiplicative group \(F^\times\) of a finite field is cyclic.
Proved that every element of a finite field of order \(q:=p^d\) is a zero of \(x^q-x\).
Along the way, we defined separable polynomials and proved that \(f\) is separable if and only if \({\rm gcd}(f,f')=1\).
Proved that for every prime power \(q:=p^d\), a splitting field of \(x^q-x\) over \(\mathbb{F}_p\) is a finite field of order \(q\).
Stated that such a field is unique up to an isomorphism. We will prove this in the next lecture.
-
3/9 (25)
Proved that a finite field of order \(p^d\) is unique up to an isomorphism, and it is denoted by \(\mathbb{F}_{p^d}\).
Recall the definition of a separable polynomial and pointed out that if \(f\) is separable and \(g|f\), then \(g\) is separable.
Proved that if \(f\in F[x]\) is irreducible and \(F\) is of characteristic zero, then \(f\) is separable.
Proved that if \(F\) is a field of characteristic \(p>0\) and \(f\in F[x]\) is irreducible, then there exist a unique separable irreducible polynomial \(f_{\rm sep}(x)\in F[x]\) and non-negative integer \(s\) such that
\(f(x)=f_{\rm sep}(x^{p^s})\).
Deduced that if either \({\rm Char}(F)=0\) or \({\rm Char}(F)=p>0\) and \(F^p=F\), then every irreducible polynomial is separable.
Mentioned that if \({\rm Char}(F)=p>0\) and \(a\in F\setminus F^p\), then \(x^p-a\) is irrducible but not separable.
Proved that if \(E/F\) and \(E'/F'\) are field extensions, \(\theta\colon F\to F'\) is a field isomorphism, and \(\alpha\in E\) is algebraic over \(F\), then the following is a bijection
\(
{\rm Embed}_{\theta}(F[\alpha],E')\to V_{m_{\alpha;F}^\theta}(E'),\quad \widehat{\theta}\mapsto \widehat{\theta}(\alpha).
\)
The inverse map is denoted by \(\alpha'\mapsto \widetilde{\theta}_{\alpha'}\).
Proved that if \(E/F\) and \(E'/F'\) are two field extensions, \(E/F\) is a finite extension, and \(\theta\colon F\to F'\) is a field isomorphism, then \(|{\rm Embed}_{\theta}(E,E')|\leq [E:F]\).
Pointed out that if equality holds, then for every \(\alpha\in E\),
\(\deg m_{\alpha;F}=|V_{m_{\alpha;F}^\theta}(E')|\) which means (1) \(m_{\alpha;F}\) is separable, and (2) all zeros of \(m_{\alpha;F}^\theta\) are in \(E'\).
-
3/11 (26)
Defined separable, normal, and Galois extensions.
Proved that if \(|{\rm Aut}_F(E)|=[E:F]\), then \(E/F\) is a Galois extension.
Proved that if \(\theta\colon F\to F'\) is a field isomorphism, \(E\) is a splitting field of \(f\in F[x]\) over \(F\), \(E'\) is a splitting field of \(f^{\theta}\) over \(F'\),
and all the irreducible factors of \(f\) in \(F[x]\) are separable, then \(|{\rm Isom}_{\theta}(E,E')|=[E:F]\).
Along the way, we proved that product of distinct separable irreducible polynomials is separable.
We also discussed that if \(E\) is a splitting field of \(f\) over \(F\), then it is also a splitting field of the square-free part of \(f\) over \(F\).
Proved that for a finite extension \(E/F\), the following are equivalent:
- \(E\) is a splitting field of an irreducible-separable polynomial \(f\in F[x]\) (this means all the irreducible factors are separable) over \(F\).
- \(|{\rm Aut}_F(E)|=[E:F]\).
- \(E/F\) is a Galois extension.
-
3/13 (27)
For a finite extension \(E/F\), we defined the following maps: \(\Phi\colon {\rm Sub}({\rm Aut}_F(E))\to {\rm Int}(E/F), \quad \Phi(G):=E^G\) and
\(\Psi\colon {\rm Int}(E/F) \to {\rm Sub}({\rm Aut}_F(E)),\quad \Psi(K):={\rm Aut}_K(E)\), where \({\rm Sub}({\rm Aut}_F(E))\) is the set of all subgroups of \({\rm Aut}_F(E)\) and
\({\rm Int}(E/F)\) is the set of all intermediate subfields of \(E/F\).
Proved that these are well-defined order-reversing maps.
Showed that \(\Phi\circ \Psi(K)\supseteq K\) and \(\Psi\circ \Phi(G)\supseteq G\). Used this to show that \([E:E^G]\geq |G|\).
Proved that if \(G\subseteq {\rm Aut}_F(E)\), \(V\subseteq E^m\) is a non-zero \(G\)-invariant subspace of \(E^m\), then \(V^G\neq 0\). Used this result to prove the following:
if \(G\) is a subgroup of \({\rm Aut}_F(E)\), then the following holds:
- \(E/{\rm Fix}(G)\) is a Galois extension.
- \(G={\rm Aut}_{{\rm Fix}(G)}(E)\).
This shows that \(\Psi\circ \Phi={\rm id}\).
Finally we proved that if \(E/K\) is a (finite) Galois extension, then \({\rm Fix}({\rm Aut}_K(E))=K\). This means that \(\Phi\circ \Psi(K)=K\) if \(E/K\) is a Galois extension.
Altogether, we have that \(\Phi\) and \(\Psi\) give us order-reversing bijections between \({\rm Sub}({\rm Aut}_F(E))\) and \( \{K\in {\rm Int}(E/F)\mid E/K \text{ is Galois}\}\).
|
ucsd 
edu)