Book
Here is a list of good textbooks that you can consult for reference.
I will be posting my lecture notes in this page, and I hope you would find them
useful. I will not cover the topics in the same order as they are presented in
the main text book; and we will learn some topics that are not covered in the main book.
The main book has an excellent collection of problems and examples; and I encourage you
to go over them and ask me or Francois if you have difficulty solving any of them.
 Dummit and Foote, Abstract Algebra third edition. (The main text book).
 Isaacs, Algebra .
 Hungerford, Algebra .
 Morandi, Field and Galois theory . (For the second half of the course).

Since some parts of field theory will be covered in 200C by Professor Bucur, I am summarizing the precise topics that I covered during the lectures.
You can review them from my notes or your book.

Suppose \(E/F\) is a field extension and \(\alpha\in E\) is algebraic over \(F\). Then the following statements hold:

There is a unique monic polynomial \(m_{\alpha,F}(x)\in F[x]\) such that for all \(f\in F[x]\), \(f(\alpha)=0\) if and only if \(m_{\alpha,F}f\).
The polynomial \(m_{\alpha,F}\) is called the minimal polynomial of \(\alpha\) over \(F\).

The minimal polynomial \(m_{\alpha,F}\) is irredcuible in \(F[x]\); moreover if \(p(x)\) is irreducible in \(F[x]\) and \(p(\alpha)=0\), then \(p(x)=cm_{\alpha,F}(x)\) for some
\(c\in F\).

\(F[\alpha]\simeq F[x]/\langle m_{\alpha,F}\rangle\), \(F[\alpha]\) is a field, and \((1,\alpha,\ldots,\alpha^{d1})\) is an \(F\)basis of \(F[\alpha]\),
where \(d=\deg m_{\alpha,F}\). In particular, \([F[\alpha]:F]=\deg m_{\alpha,F}\).

For every irreducible polynomial \(f\in F[x]\), there exists a pair \((E,\alpha)\) such that
 \(E/F\) is a field extension and \(E=F[\alpha]\).
 \(f(\alpha)=0\).

Suppose \(\theta:F\to F'\) is a field isomorphism and \(f\in F[x]\) is irreducible. Suppose \((E,\alpha)\) and \((E',\alpha')\) satisfy the following properties: \(E/F, E'/F'\)
are field extensions, \(E=F[\alpha], E'=F'[\alpha']\), and \(f(\alpha)=f^\theta(\alpha)=0\). Then there is a \(\theta\)isomorphism \(\widehat{\theta}:E\to E'\) such that
\(\widehat{\theta}(\alpha)=\alpha'\).

For every polynomial \(f\in F[x]\), there exists a splitting field \(E\) of \(f\) over \(F\).

Suppose \(\theta:F\to F'\) is a field isomorphism, \(f\in F[x]\), \(E\) is a splitting field of \(f\) over \(F\), and \(E'\) is a splitting field of \(f^\theta\) over \(F'\).
Then there exists an isomorphism \(\widehat{\theta}:E\to E'\) which is an extension of \(\theta\). In particular, a splitting field of \(f\) over \(F\) is unique up to
an \(F\)isomorphism.

For every prime \(p\) and positive integer \(n\), there is a unique up to an isomorphism field of order \(p^n\). It is denoted by \(\mathbb{F}_{p^n}\). Field \(\mathbb{F}_{p^n}\) is a
splitting field of \(x^{p^n}x\) over \(\mathbb{Z}/p\mathbb{Z}\). We have \(x^{p^n}x=\prod_{\alpha\in \mathbb{F}_{p^n}} (x\alpha).\)

A polynomial \(f\in F[x]\) is called separable if \(f\) does not have multiple zeros in a splitting field of \(f\) over \(F\). We proved that \(f\) is separable if and only if
\(\gcd(f,f')=1\). An irreducible polynomial \(f\) is separable if and only if \(f'\neq 0\); in particular, over a field of characteristic zero, every irreducible polynomial is
separable. If \(F\) is a field of characteristic \(p>0\), then for every \(f\in F[x]\), there exists a separable polynomial \(g\in F[x]\) such that \(f(x)=g(x^{p^k})\) for some
nonnegative integer \(k\).

Suppose \(K\) is an intermediate field of \(E/F\). Then \([E:F]=[E:K][K:F]\).

Suppose \(\theta:F\to F'\) is a field isomorphism, \(f\in F[x]\), \(E\) is a splitting field of \(f\) over \(F\), and \(E'\) is a splitting field of \(f^\theta\) over \(F'\).
Then \({\rm Isom}_{\theta}(E,E')\leq [E:F]\) and the equality holds precisely when all the irredcuible factors of \(f\) in \(F[x]\) are separable. In particular, if \(E\) is
a splitting field of \(f\) over \(F\), then \({\rm Aut}(E/F)\leq [E:F]\) and equality holds precisely when all the irredcuible factors of \(f\) in \(F[x]\) are separable.

Suppose \(F\) and \(F'\) are two subfields of \(E\), \(\theta:F\to F'\) is an isomorphism, and \([E:F]< \infty\). Then \({\rm Isom}_\theta(E,E)\leq [E:F]\).

An algebraic extension \(E/F\) is called

a normal extension if for all \(\alpha\in E\), \(m_{\alpha,F}\) factors into linear terms over \(E\),

a separable extension if for all \(\alpha\in E\), \(m_{\alpha,F}\) is separable.

a Galois extension if it is both normal and separable.

Suppose \(E/F\) is a finite extension. Then the following statements are equivalent.
 There exists a polynomial \(f\in F[x]\) with separable irreducible factors such that \(E\) is a splitting field of \(f\) over \(F\).
 \({\rm Aut}(E/F)=[E:F]\).
 \(E/F\) is a Galois extension.

Suppose \(G\) is a subgroup of the group of automorphisms of a field \(E\). Consider the action of \(G\) on \(E^n\). Suppose \(V\) is a nonzero subspace of \(E^n\)
which is \(G\)invariant. Then \(V^G\) is nonzero.

Suppose \(G\) is a subgroup of the group of automorphisms of a field \(E\). Then \({\rm Fix}(G)\) is a subfield of \(E\) and \(G\subseteq {\rm Aut}(E/{\rm Fix}(G))\).

Suppose \(G\) is a finite subgroup of the group of automorphisms of a field \(E\). Then \([E:{\rm Fix}(G)]\leq G\).

Prove that \({\rm Aut}_{\mathbb{F}_p}(\mathbb{F}_{p^n})=\langle \sigma_p\rangle\) where \(\sigma_p:\mathbb{F}_{p^n}\to \mathbb{F}_{p^n}, \sigma_p(a):=a^p\)
is the Frobenius endomorphism.

Suppose \(E/F\) is a Galois extension and \(K\) is an intermediate subfield of \(E/F\). Then \(E/K\) is a Galois extension.

Suppose \(E/F\) is a finite Galois extension and \(K\) is an intermediate subfield of \(E/F\). Then the following properties are equivalent:

\(K/F\) is a normal extension.

For every \(\theta\in {\rm Aut}_F(E)\), \(\theta(K)=K\).

Suppose \(E/F\) is a finite Galois extension and \(K\) is an intermediate subfield of \(E/F\) such that \(K/F\) is a normal extension. Then the restriction induces
a group homomorphism \(r_{E/K}:{\rm Aut}_F(E)\to {\rm Aut}_F(K)\) and the following is a SES:
\(
1\to {\rm Aut}_K(E)\to {\rm Aut}_F(E) \xrightarrow{r_{E/K}}{\rm Aut}_F(K)\to 1.
\)

When \(E/F\) is a Galois extension, we write \({\rm Gal}(E/F)\) instead of \({\rm Aut}_F(E)\).
 The fundamental theorem of Galois theory. Suppose \(E/F\) is a finite Galois extension. Let \({\rm Int}(E/F)\) be the set of all the intermediate subfields of \(E/F\) And
\({\rm Sub}({\rm Gal}(E/F))\) be the set of all the subgroups of \({\rm Gal}(E/F)\). Let
\(
\Phi:{\rm Int}(E/F)\to {\rm Sub}({\rm Gal}(E/F)), \Phi(K):={\rm Gal}(E/K),
\)
and
\(
\Psi:{\rm Sub}({\rm Gal}(E/F)) \to {\rm Int}(E/F), \Psi(G):={\rm Fix}(G).
\)
Then the following statements hold.

\(\Phi\) and \(\Psi\) are inverse of each other; this means \({\rm Gal}(E/{\rm Fix}(G))=G\) and \({\rm Fix}({\rm Gal}(E/K))=K\).

\(\Phi(K)\) is a normal subgroup of \({\rm Gal}(E/F)\) if and only if \(K/F\) is a normal extension.
Consequently, \(\Psi(G)/F\) is a normal extension if and only if \(G\) is a normal subgroup \({\rm Gal}(E/F)\).

\(\Phi\) and \(\Psi\) are reverse ordering isomorphisms between the POSets \({\rm Int}(E/F)\) and \({\rm Sub}({\rm Gal}(E/F))\).

Suppose \(F\) is a field with an element \(\zeta\) of multiplicative order \(n\). Suppose either the charactersitic of \(F\)
is zero or it does not divide \(n\). Suppose \(a\in F^\times\). Let \(E\) be a splitting field of \(x^na\) over \(F\). Then the following statements hold.

\(E/F\) is a finite Galois extension and
\(E=F[\sqrt[n]{a}]\) where \(\sqrt[n]{a}\in E\) is a zero of \(x^na\).

\(x^na=\prod_{i=0}^{n1}(x\zeta^i \sqrt[n]{a})\).

\(\iota: {\rm Gal}(E/F)\to \langle \zeta\rangle,\quad \iota(\theta):=\frac{\theta(\sqrt[n]{a})}{\sqrt[n]{a}}\) is an injective group homomorphism. In particular,
\({\rm Gal}(E/F)\) is a cyclic group and its order divides \(n\).
