$\DeclareMathOperator{\Q}{Q} \DeclareMathOperator{\Aut}{Aut}$

# COURSE YEAR 2017

Module - I : April 30, 2017 – May 12, 2017, Michel Waldschmidt
Topics: Rings, Fields, The characteristic of a field, Review of polynomial rings, Factoring polynomials, Extension fields, The subring gene-rated by a subset, The subfield generated by a subset, Construction of some extension fields, Stem fields, Algebraic and transcendental elements, Transcendental numbers
[Homeworks]

• 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 $\mathbb{Q}(i)$, $\mathbb{Q}(\sqrt 2)$.

• 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.

• 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.

• 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.

• 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.