Groups, Rings and Fields (ECM2711)

Students that take this course are expected to have a strong interest in Algebra.
The level of difficulty of this course is considerably higher than that of Numbers, Symmetries and Groups (ECM1706).

There are two pieces of assessed coursework, each counts for 10%. See the ELE page for the course for hand in dates, exercises sheets and solutions.

Syllabus:

I.1. Basics about groups:
L1-3, p1-10: review of group axioms, subgroups and basic examples: symmetric groups, dihedral groups, cyclic groups;
L4-6, p6-10: homomorphism, kernel, image, isomorphism; the sign of a permutation, the alternating group.
I.2. Basics about group actions:
L6-8, p11-15: group action, examples, orbit, stabiliser, the orbits form a partition, the stabilisers are subgroups, Orbit-Stabiliser Theorem;
L8-12, p15-23: Orbit Counting Lemma and examples (colourings), conjugacy classes, centre of a group, centraliser,
cycle type, the conjugacy classes of $S_n$ are labelled by partitions of $n$, direct product of groups, conjugacy classes and centralisers in $S_5$ and $A_5$.
I.3. Further results about groups:
L13-16, p24-30: left and right cosets, normal subgroup, quotient group, the canonical homomorphism, First Isomorphism Theorem,
L16-18, p30-34: Sylow's three theorems and applications.

II.1. Basics about rings:
L19-21, p1-4: definition of a ring, commutative ring, Examples: $\mathbb Z, \mathbb Q, \mathbb R, \mathbb C$, matrices/linear transformations, polynomials, functions, $\mathbb{Z}/n\mathbb{Z}$,
unit, field, zero divisor, integral domain, degree and leading coefficient of a polynomial, subring, field of fractions of an integral domain;
L21-23, p4-7: ideal, homomorphism, kernel, image, the characteristic of a ring, the $p$-th power map in characteristic $p$,
quotient ring, the canonical ring homomorphism, First Isomorphism Theorem for rings.
II.2. Principal ideal domains & unique factorisation:
L23-25, p7-10: Division with remainder for polynomials, ideal generated by given ring elements, principal ideal domain (PID),
division and equivalence ( | and $\sim$), monic polynomial, gcd and lcm, Euclid's gcd algorithm for integers and polynomials ($\mathbb{Z}$ and $F[X]$ );
L26-28, p10-13: prime ideal, maximal ideal, irreducible and prime elements, unique factorisation domain (UFD),
in a PID every irreducible element is prime and every nonzero prime ideal is maximal, every PID is a UFD, gcd and lcm for any UFD;
L29-30, p13-15: Irreducibility criteria for $\mathbb Q[X]$: primitive polynomials, Gauss's Lemma and its corollaries, Eisenstein's criterion for irreducibility.
II.3. Algebraic numbers and minimal polynomials:
L30-33, p16-18: algebraic numbers, minimal polynomial, examples (including roots of unity), the theorem on simple field extensions $F[\alpha]=F(\alpha)$,
transcendental numbers.

Prerequisites:

Numbers, Symmetries and Groups (ECM1706)
Vectors and Matrices (ECM1701)

The literature for this course:

J.R. Durbin, Modern Algebra: An Introduction, sixth edition, John Wiley & Sons, 2009.
D.A.R. Wallace, Groups, Rings and Fields, Springer-Verlag London, 1998.
P.J. Cameron, Introduction to Algebra, second edition, Oxford University Press, 2008.
The algebra books by P.M. Cohn, S. Lang and B.L. van der Waerden.

The above books are only for supplementary reading. I will also make hand written lecture notes available.

Learning outcomes

On successful completion of ECM2711, students will be able to:

Give the definition of: subgroup, cyclic subgroup, normal subgroup, quotient group, homomorphism, kernel, image, isomorphism,
group action, orbit, stabiliser, the sign of a permutation, the alternating group, conjugacy class, cycle type.
Prove elementary properties of above concepts. Check (in easy cases) whether or not a given subset of a group is a (normal) subgroup.
Calculate with permutations (e.g. determine the the disjoint cycle form of a permutation). State and apply the Orbit-Stabiliser Theorem,
the Orbit Counting Lemma and Sylow's Theorems.
Give the definition of: ring, commutative ring, unit, zero divisor, subring, ideal, integral domain, field, field of fractions of an integral domain,
ring homomorphism, prime ideal, maximal ideal, the characteristic of a ring, irreducible/prime element, principal ideal domain,
unique factorisation domain, primitive polynomial, minimal polynomial.
Prove elementary properties of above concepts. Check (in easy cases) whether or not a given set with operations is a ring and whether or not
a given subset of a ring is a subring/ideal/prime ideal/maximal ideal.
Calculate the gcd of two polynomials using Euclid's algorithm. State and apply the First Isomorphism Theorem for rings, Gauss's lemma and
Eisenstein's criterion for irreducibility.

Remarks about the lectures

For the basics about sets and functions, see the pdf documents on sets and functions in the section Further Resources on the ELE page for this course.
The lecture on 30-9-2013.
The order of an element $x\in G$ is defined as the smallest integer $r>0$ such that $x^r=e$ if such an $r$ exists and $\infty$ otherwise.
The order of a group is the number of elements it contains.
We have seen that that the order of $x\in G$ is equal to the order of the subgroup $\langle x\rangle$ it generates.
So, by Lagrange's Theorem, the order of $x\in G$ divides the order of $G$.
Recall that Lagrange's Theorem says that the order of a subgroup of a group $G$ always divides the order of $G$.
The lecture on 7-10-2013.
Another way to state the result that $\{{\rm Orb}(x)\,|\,x\in X\}$ is a partition of a set $X$ on which the group $G$ acts is to say that the relation
$x\sim y\stackrel{\rm def}{\Longleftrightarrow} \exists_{g\in G}\,x=g\cdot y$
is an equivalence relation on $X$.
For the definitions of partition and equivalence relation see one of the pdf documents on sets and functions in the section Further Resources
on the ELE page for this course.
The lecture on 18-10-2013.
Recall that when a group $G$ acts on a set $X$ I said that $x,y\in X$ are conjugate if $\exists_{g\in G}\,x=g\cdot y$.
If I say that $x,y\in G$ are conjugate without specifying under which action of which group, then I always mean under the conjugation action of $G$ on itself.
So $x,y\in G$ are conjugate if (and only if) $\exists_{g\in G}\,x=gyg^{-1}$.
The lecture on 25-10-2013.
The issue with the well-definedness of the multiplication on $G/H$ and the homomorphism $\overline{\varphi}$ of the First Isomorphism Theorem is that the elements
on which these maps are defined are expressions involving a “parameter”, and this parameter is used in the definition of the function.
In the first case the expressions are $(g_1H,g_2H)$ involving the parameter $(g_1,g_2)$, in the second case the expressions are $gK$ involving the parameter $g$.
An example of totally different nature may make things more clear. Let $Q=\{n^2\,|\,n\in\mathbb Z\}\subset\mathbb Z$ be the set of squares of integers.
Then we could attempt to define a map $f:Q\mathbb\to\mathbb Z$ by $f(n^2)=n+1$. Then $1^2=(-1)^2$, but $1+1\ne-1+1$. So $f$ is not well-defined.
If we define $f:Q\mathbb\to\mathbb Z$ by $f(n^2)=|n|+1$, then $f$ is well-defined.
The lecture on 4-11-2013.
When I defined the product of two polynomials I wrote $f(X)g(X)=\sum_{i=0}^{mn}c_iX^i$, where $c_i=\cdots$.
This should have been $f(X)g(X)=\sum_{i=0}^{m+n}c_iX^i$, where $c_i=\cdots$.
The lecture on 14-11-2013.
Note that $R/\{0\}\cong R$ (in the case of groups we have $G/\{1\}\cong G$).
The lecture on 22-11-2013.
In the proof of the existence part of the theorem that every PID is a UFD I should probably have expressed the idea a bit more clear.
The idea is that If $a\in R\setminus\{0\}$ is not irreducible, then you can write $a=bc$ where $b$ and $c$ are non-units.
If $b$ is not irreducible then, you can write it as a product $b=b_1b_2$ where $b_1,b_2$ are non-units. So then $a=bc=b_1b_2c$.
The same applies of course to $c$ and to $b_1$ and $b_2$. So we can continue replacing certain non-irreducible factors in a factorisation of $a$
by a product of two non-units until all the factors are irreducible. Note, however, that it is not clear why this process should terminate.
This is why I used the absolute value in the case of $\mathbb Z$ and the degree in the case of $F[X]$.
The lecture on 29-11-2013.
I was a bit distracted when I gave the proof of Eisenstein's criterion.
The argument was almost complete: We have $f(X)=g(X)h(X)$ (*), where $g(X)$ has degree l and $h(X)$ has degree $m$. So $l+m=n$.
Now we had $a_nX^n=g(X)h(X)$ modulo $p$ (**). So $g(X)$ and $h(X)$ are modulo $p$ scalar multiples of powers of $X$.
But $h(X)$ has a constant term which is nonzero mod $p$, so modulo $p$ it has to be a constant.
Here I got a bit stuck. I wanted that $h(X)$ itself is a constant. A degree argument gives this:
By (**) $g(X)$ has to be a multiple of $X^n$ modulo $p$, so its degree modulo $p$ is $n$, but then its (ordinary) degree $l$ is also $n$.
So $m=0$, i.e. $h(X)$ is a constant.
Another way to get there goes as follows. From (*) and the fact that $a_n$ is nonzero mod $p$ it follows that $b_lc_m=a_n\not\equiv0$ mod $p$, so $b_l,c_m\not\equiv0$ mod $p$.
But we also know that $c_0\not\equiv0$ mod $p$ and that $h(X)$ is modulo $p$ a scalar multiple of a power of $X$. So we must have $m=0$, i.e. $h(X)$ is a constant.
The lecture on 2-12-2013.
Standard examples of complex numbers that are not algebraic numbers (such numbers are called transcendental (over $\mathbb Q$)) are $e$ and $\pi$.
The proof that these are indeed not algebraic numbers is beyond the scope of this course. It is, however, not to hard to show that the set of algebraic
numbers is countable. Since $\mathbb C$ is uncountable, this shows at least that there are transcendental numbers, uncountably many in fact.
The lecture on 6-12-2013.
The final remark that I made about finding all the roots of a polynomial $X^n-a$, $a\in\mathbb C\setminus\{0\}$ can be used to find factorisations over the real or rational
numbers by first factorising over the complex numbers, i.e. finding all the roots. For example, you can find a factorisation of $X^4+4$ over $\mathbb R$ (or $\mathbb Q$) by first
finding the root $\sqrt{2}\zeta_8$, where $\zeta_8=e^{2\pi i/8}$ and then finding the others using the remark from the lecture. Finally you can then find a real factorisation by
taking factors corresponding to complex conjugate roots together.