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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).
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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.
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Suppose \(E/F\) is a field extension and \(\alpha\in E\) is algebraic over \(F\). Then the following statements hold:
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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 \(F\).
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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
\(c\in F\).
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\(F[\alpha]\simeq F[x]/\langle m_{\alpha,F}\rangle\), \(F[\alpha]\) is a field, and \((1,\alpha,\ldots,\alpha^{d-1})\) is an \(F\)-basis of \(F[\alpha]\),
where \(d=\deg m_{\alpha,F}\). In particular, \([F[\alpha]:F]=\deg m_{\alpha,F}\).
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For every irreducible polynomial \(f\in F[x]\), there exists a pair \((E,\alpha)\) such that
- \(E/F\) is a field extension and \(E=F[\alpha]\).
- \(f(\alpha)=0\).
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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
\(\widehat{\theta}(\alpha)=\alpha'\).
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For every polynomial \(f\in F[x]\), there exists a splitting field \(E\) of \(f\) over \(F\).
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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.
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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 \(x^{p^n}-x=\prod_{\alpha\in \mathbb{F}_{p^n}} (x-\alpha).\)
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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
non-negative integer \(k\).
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Suppose \(K\) is an intermediate field of \(E/F\). Then \([E:F]=[E:K][K:F]\).
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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.
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Suppose \(F\) and \(F'\) are two subfields of \(E\), \(\theta:F\to F'\) is an isomorphism, and \([E:F]< \infty\). Then \(|{\rm Isom}_\theta(E,E)|\leq [E:F]\).
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An algebraic extension \(E/F\) is called
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a normal extension if for all \(\alpha\in E\), \(m_{\alpha,F}\) factors into linear terms over \(E\),
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a separable extension if for all \(\alpha\in E\), \(m_{\alpha,F}\) is separable.
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a Galois extension if it is both normal and separable.
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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.
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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 non-zero subspace of \(E^n\)
which is \(G\)-invariant. Then \(V^G\) is non-zero.
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Suppose \(G\) is a subgroup of the group of automorphisms of a field \(E\). Then \({\rm Fix}(G)\) is a subfield of \(E\) and \(G\subseteq {\rm Aut}(E/{\rm Fix}(G))\).
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Suppose \(G\) is a finite subgroup of the group of automorphisms of a field \(E\). Then \([E:{\rm Fix}(G)]\leq |G|\).
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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.
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Suppose \(E/F\) is a Galois extension and \(K\) is an intermediate subfield of \(E/F\). Then \(E/K\) is a Galois extension.
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Suppose \(E/F\) is a finite Galois extension and \(K\) is an intermediate subfield of \(E/F\). Then the following properties are equivalent:
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\(K/F\) is a normal extension.
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For every \(\theta\in {\rm Aut}_F(E)\), \(\theta(K)=K\).
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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:
\(
1\to {\rm Aut}_K(E)\to {\rm Aut}_F(E) \xrightarrow{r_{E/K}}{\rm Aut}_F(K)\to 1.
\)
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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.
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\(\Phi\) and \(\Psi\) are inverse of each other; this means \({\rm Gal}(E/{\rm Fix}(G))=G\) and \({\rm Fix}({\rm Gal}(E/K))=K\).
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\(\Phi(K)\) 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 \({\rm Gal}(E/F)\).
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\(\Phi\) and \(\Psi\) are reverse ordering isomorphisms between the POSets \({\rm Int}(E/F)\) and \({\rm Sub}({\rm Gal}(E/F))\).
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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 \(x^n-a\) over \(F\). Then the following statements hold.
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\(E/F\) is a finite Galois extension and
\(E=F[\sqrt[n]{a}]\) where \(\sqrt[n]{a}\in E\) is a zero of \(x^n-a\).
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\(x^n-a=\prod_{i=0}^{n-1}(x-\zeta^i \sqrt[n]{a})\).
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\(\iota: {\rm Gal}(E/F)\to \langle \zeta\rangle,\quad \iota(\theta):=\frac{\theta(\sqrt[n]{a})}{\sqrt[n]{a}}\) is an injective group homomorphism. In particular,
\({\rm Gal}(E/F)\) is a cyclic group and its order divides \(n\).
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