Nepal Algebra Project (NAP) / नेपाल बीज-गणित परियोजना


Module - I : May 08, 2016 – May 20, 2016, Roger Wiegand and Sylvia Wiegand
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
Brief review of basic notions (groups, rings, and fields): (also download from here)

  • Characteristic of a field, "Freshman's Dream": $\left(a+ b\right)^p$ = $a^p$ + $b^p$
  • Division Algorithm (with proof); $x-c \mid f(x)$ $\iff$ $f \left(a \right)$= 0
  • This includes items 1.1-1.4, 1.5, 1.6, 1.7, 1.9, 1.11 $F[x]$ is a PID; precise definition of UFD. If degree $f\left(x\right)\le$ 3 then $f\left(x \right)$ is irreducible $\iff f \left(x \right)$ has no roots; fails for degree 4

Discussion of irreducibility in $\mathbb {Z}[X]$ vs in $\mathbb{Q}[X]$: (also download from here)
  • Testing for irreducibility in $\mathbb{Z}[X]$ by checking irreducibility in $\mathbb{F}_p[X]$; proof of why it works, pitfalls of misinterpreting, why $p$ must not divide leading coefficient. Explained what Milne means by "factors non-trivially" (not quite the same, over $\mathbb{Z}$, as being reducible). Examples, of a polynomial that can be shown to be irreducible by going mod 2 and of another polynomial, $X^4-10X^2+1$ in $\mathbb{Z}[X]$, that factors non-trivially modulo every prime, even though it is irreducible (proof of irreducibility assigned in Problem Set 1).
  • Finding a polynomial in $\mathbb{Z}[X]$ with $\sqrt 2+\sqrt 3$ as a root (the polynomial above, coincidentally).
  • Gauss' Lemma 1.13, proved with a more rigorous version of Milne's proof. (Choose appropriate positive rational multiples of $g(x)$ and $h(x)$ so as to minimize their integral product. then show that this minimal product must be 1.)
  • Vector spaces: Discussed how $F[X]$, and extension fields $K$ of $F$ are vector spaces over $F$.
  • Proposition 1.8: proved that for $R$ a PID, gcd's exist and can be written as a combination of the generators. Discussed the Euclidean algorithm method of finding $d=\gcd(a,b)$ and r,s with $ra +sb=d$.
  • Stated and proved one case of the $\frac{2}{3}$ Lemma ("two-out of three lemma"): If $a,b,c,d\in R$, a commutative ring, with $d\ne 0$ and $a \pm b=c$, and two of a, b, c are multiples of d, then so is the third.
On Tuesday May 17 2016 (also download from here)
  • Discussed extension fields, notation, as on page 13 of Milne, including ``$F$-homomorphism'', and examples.
  • We added the fact, which we proved later, that, if $\alpha$ and $\beta$ are roots of the same irreducible polynomial there exists an $F$--homomorphism from $F\big[ \alpha \big] $ to $F \big[ \beta \big]$
  • Proved Proposition 1.20 Multiplication of degrees, part 2. (Will go over part 1 in lecture 4 today.)
  • Proved Lemma 1.2.3: For $F\subseteq R$, where $F$ is a field and $R$ is an integral domain that has finite dimension over $F$ as a vector space, $R$ is a field.
  • Discussed constructing an extension field where an irreducible polynomial has a root. Proved it is a field.
  • Discussed examples such as building the complex numbers from the reals.
  • Proved the finiteness Theorem: For $F \subseteq E \subseteq K$ fields, these are equivalent:
    1. $E/F$ is finite (vector space dimension).
    2. $E/F$ is finitely generated as a field extension and is algebraic over $F$.
    3. There exist algebraic elements $\alpha_1$, $\alpha_2$, $\dots$, $\alpha_n$ of $E$, such that $E=F(\alpha_1, \alpha_2,$ $\dots, \alpha_n)$.
On Wednesday May 18 2016 (also download from here)
  • Reviewed the Finite-algebraic Theorem from Lecture 5: For $F$ subseteq $E$ fields, these are equivalent:
    1. $E/F$ is finite (i.e. finite vector space dimension).
    2. $E/F$ is finitely generated as a field extension and $E$ is algebraic over $F$.
    3. There exist algebraic elements $\alpha_1$, $\alpha_2$, $\dots$, $\alpha_n$ of $E$, such that $E=F(\alpha_1, \alpha_2, \dots,$ $\alpha_n)$.
  • Stated and proved the Corollary: Let $K/F$ be a field extension and let $\overline F$ be the set of elements of $K$ that are algebraic over $F$. Then $\overline F$ is a subfield of $K$ with $F \subset \overline F \subset K$. The field $\overline F$ is called the algebraic closure of $F \in K$.
  • Discussed the set of algebraic numbers in the complex numbers $\mathbb {C}$ over the rational numbers $\mathbb {Q}$.
  • Showed that for $\alpha=\sqrt 2 + \sqrt 3$, $\mathbb{Q}(\alpha)= \mathbb{Q} (\sqrt2, \sqrt 3)$. Later showed that $\sqrt 3 \notin \mathbb{Q} (\sqrt2) $ and consequently that $ \big[ \mathbb{Q}(\alpha): \mathbb{Q} \big]=4$ (By Eisenstein $x^2-2$, and $x^2-3$ are irreducible over $\mathbb{Q}$ and so $ \sqrt 2, \sqrt 3 \notin \mathbb {Q}$).
  • Discussed an example of the Louisville numbers (with big gaps in the decimal expansion): $\sum_{i=1}^\infty 1/10^{i!}$. These are transcendental and it is easier to show they are transcendental than other transcendental numbers (see book).
  • Discussed countable and uncountable cardinalities.
  • Stated and proved the Theorem : For $K/F$ a field extension with $F$ countable, the set of elements of $K$ that are algebraic over $F$ is countable.
  • To prove this, stated and outlined/picture-proved $N$ lemmas:
    For $X, X_n, Y$ sets
    1. $X,Y$ countable $\implies X \times Y$ countable;
    2. $X$ countable and an injection takes $Y$ into X $\implies Y$ is countable;
    3. $X$ countable and a surjection takes $ X \to Y \implies Y$ countable;
    4. $Y$ a countable union of countable sets $X_n \implies Y$ is countable;
    5. $F$ a countable field and $V$ a vector space over $F$ with a countable basis $\implies V$ is countable;
    6. $\vdots$
    7. $N$. If $X$ is countable and $Y \to X$ is countable-to-one, then $Y$ is countable.
  • Corollary: In the complex numbers $\mathbb{C}$, the subfield of elements that are algebraic over the rationals $\mathbb{Q}$ is countable. Elements of this set are called algebraic numbers.
  • Proved part $1$ of Proposition~1.20 Multiplication of degree (part 2 was proved in Lecture 5).
  • Proved the following tw statements:
    Proposition: (algebraic extensions are transitive): If $F \subseteq K \subseteq L$ are fields with $K/F$ algebraic and $L/K$ algebraic, then $L/F$ is algebraic.
    Proposition: If $F\subseteq K \subseteq L$ are fields and $L=F[\alpha]$ where alpha is algebraic, then the minimal polynomial for $\alpha$ over $K$ divides the minimal polynomial for $\alpha$ over $F$.
  • Worked two problems:
    • Show if $[L:F]=p$, a prime integer, then there are no fields $K$ properly between $F$ and $L$.
    • If $F \subseteq L$ is algebraic and $T$ is an integral domain with $F \subseteq T \subseteq L$, prove that $T$ is a field.
    Another problem the students might like to try for extra practice is Exercise~1-5 in Milne.