Topics:
Rings, Fields, The characteristic of a field, Review of polynomial rings, Factoring polynomials, Extension fields, The subring generated by a subset, The subfield generated by a subset, Construction of some extension fields, Stem fields, Algebraic and transcendental elements, Transcendental numbers
[Homeworks]
Tuesday, May 2, 2017
(also download from here)
 Introduction.

Starting from the set $\mathbb{N}=\{0,1,2,\dots\}$ of natural numbers, construction of the ring $\mathbb{Z}$ of rational integers, next of the field $\mathbb{Q}$ of rational numbers (as a quotient), of the field $\mathbb{R}$ of rational numbers (no detail), and then construction of $\mathbb{C}$ as a quotient $\mathbb{R}[X]/(X^2+1)$.
 Complex conjugation.

Rings, integral domains, group of units. Units of $\mathbb{R}[X]$ when $\mathbb{R}$ is a domain.
 Definition of irreducible polynomials over a field.
 Property (without proof): $\mathbb{C}$ is algebraically closed.
 Exercises:
 Which are the irreducible polynomials over $\mathbb{R}$?
 The fields $\mathbb{Q}(i)$, $\mathbb{Q}(\sqrt 2)$.
Wednesday May 3, 2017
(also download from here)

Solution of the exercises of the first course:
 irreducible polynomials over $\mathbb{R}$

$\sqrt{2}$ as a limit of a Cauchy sequence of rational numbers.
 How to prove that any subgroup of $\mathbb{Z}$ is of the form $n\mathbb{Z}$?
 Euclidean algorithm for $\mathbb{Z}$; gcd, Bézout.
 Homomorphisms, image (subring), kernel (ideal).

How to prove that any ideal ${\mathcal{ I} }\not =R$ in a domain $R$ is the kernel of a ring homomorphism?
 Prime ideals, maximal ideals. Examples.
 Exercises.

The quotient rings $\mathbb{Z}[X]/(X)$, $\mathbb{Z}[X]/(2)$, $K[X,Y]/(Y)$.

The characteristic of a field; the smallest subfield. The Frobenius endomorphism.

The ring of polynomials over a domain $\mathbb{R}$. Euclidean algorithm for the ring of polynomials in one variable over a field; division by a monic polynomial over a domain.
Thursday May 4, 2017
(also download from here)

Subring generated by a subset. Subfield generated by a subset. Ideal generated by a subset.
 Algebraic vs transcendental elements over a field.
 Finite extension, algebraic extension. Degree of an extension.
 Simple extension.
 Irreducible polynomial of an algebraic number.
 If $\alpha$ is algebraic over $F$, then $F(\alpha)=F[\alpha]$.
 Frobenius endomorphism in characteristic non zero.
Friday May 5, 2017
(also download from here)
 Multiplicativity of degrees of finite extensions.
 Finitely generated extensions.
 Composite of extensions.
 Stem field. Unicity up to $F$  isomorphism. Conjugates of an algebraic element over a field.
Sunday May 7, 2017
(also download from here)
 Factoring a polynomial over $\mathbb{Q}$.

Gauss Lemma. Proof as follows:

Definition of primitive polynomials in $\mathbb{Z}[X]$: the gcd of the coefficients is $1$.
 for any nonzero polynomial $g\in \mathbb{Q}[X]$, there is a unique positive rational number $c(g)$ such that $g=c(g) g_1$ where $g_1\in\mathbb{Z}[X]$ is primitive. If $g\in\mathbb{Z}[X]$, then $c(g)$ (content of $g$) is the gcd of the coefficients of $g$.

The product of two primitive polynomials in $\mathbb{Z}[X]$ is primitive:
for $p$ a prime number, use the canonical ring homomorphism $\varphi_p:\mathbb{Z}[X]\rightarrow \mathbb{Z}_p[X]$.
 Deduce $c(fg)=c(f)c(g)$.
 Proof of Gauss Lemma.
 Eisenstein Criterion.

Existence of transcendental numbers, gave the proof following Cantor of the existence of transcendental numbers but not the proof by Liouville.

Algebraically closed fields. Algebraic closure of a field.