The ELE page for Modern Algebra (ECM3731).

I will teach the second part of the course Modern Algebra called *Rings, fields and Galois theory*.

This part is divided in three sections**The literature for this course:**

J. R. Durbin, *Modern Algebra: An Introduction*, John Wiley & Sons.
P.J. Cameron, *Introduction to Algebra*, Oxford Science Publications 1998.
I. Stewart, *Galois Theory*, Chapman and Hall 2004.
The algebra books by P.M. Cohn, S. Lang and B.L. van der Waerden.

The above books are only for supplementary reading. By the end of the course I intend to make lecture notes available. They should more or less cover the contents of the lectures.

This part is divided in three sections

- I.
*Basic definitions*: definition of a ring, commutative ring, unit, zero divisors, integral domain, field, field of fractions of an integral domain.

Examples: $\mathbb Z, \mathbb Q, \mathbb R, \mathbb C$, matrices/linear transformations, polynomials, functions, $\mathbb{Z}/n\mathbb{Z}$.

Subring, ideal (generated by a set), homomorphism, kernel, image, first isomorphism theorem, prime ideal, maximal ideal.

The characteristic of a ring, the $p$-th power map in characteristic $p$. - II.
*Principal ideal domains & unique factorisation*: irreducible and prime elements.

Principal ideal domain, unique factorisation domain, Examples: $\mathbb{Z}$ and $F[X]$ (division with remainder and Euclid's gcd algorithm for polynomials).

Minimal polynomial, primitive polynomials, Gauss's Lemma, Eisenstein's criterion for irreducibility. - III.
*Field extensions and Galois theory*: field extensions, algebraic elements, algebraic extensions, finite extensions, tower law.

Splitting fields, roots of unity, normal extensions.

Isomorphisms and automorphisms of fields, fundamental theorem of Galois theory.

The above books are only for supplementary reading. By the end of the course I intend to make lecture notes available. They should more or less cover the contents of the lectures.

On successful completion of ECM3731 Part 2, students will be able to:

- Give the definition of: ring, commutative ring, unit, zero divisor, integral domain, field, subring, ideal, prime ideal, maximal ideal, the characteristic of a ring, irreducible/prime element, principal ideal domain, unique factorisation domain, minimal polynomial, algebraic/finite/normal/Galois extension.
- 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 Gauss's lemma and Eisenstein's criterion for irreducibility, determine the minimal polynomial of a field element over a subfield.
- Calculate with small field extensions and roots of unity, state and apply the tower law for field extensions.
- State the fundamental theorem of Galois theory for a finite normal extension in $\mathbb C$ and apply it to give all intermediate fields for small normal extensions in $\mathbb C$.

- The lecture on Tuesday 19-2-2013.

I wrote ${\rm End}_F(V)$ (*not ${\rm End}_F(N)$*as several people thought) for the ring of all linear maps $:V\to V$. Here “ End” stands for “ endomorphism” which is a morphism (= map which preserves structure) from a mathematical object to itself. Next lecture I hope to write more clearly with a better marker.

In the definition of zero divisor you should assume the ring to be commutative.

You should be able to do some of the exercises on Sheet 1*after the next lecture*. - The lecture on Thursday 21-2-2013.

As one of the students pointed out, in the line “ $p|p!=\binom{p}{i}i!$” , $p!$ should be replaced by $p(p-1)\cdots(p-i+1)$. So it should be “$p|p(p-1)\cdots(p-i+1)=\binom{p}{i}i!$”.

I forgot to say what an isomorphism is. For rings, as for all algebraic stuctures, it is a bijective (= 1-1 and onto) homomorphism. - The lecture on Monday 25-2-2013.

Maybe I should have reminded you of cosets: $x+I$ means $\{x+y\,|\,y\in I\}$ which is an additive version of $xN=\{xy\,|\,y\in N\}$, $N$ a (normal) subgroup of a group $G$.

So our cosets are just a special case of those in the first part of the course: we form quotients by additive subroups (recall that in an abelian group every subgroup is normal).

The difference is that we give the set of all cosets (the quotient $R/I$ of a ring $R$ by an ideal $I$) more structure: the multiplication.

This lecture was somewhat dogmatic, but quotients will be important later on in the course where we consider quotients of the polynomial ring $F[X]$ as extensions of $F$. - The lecture on Tuesday 26-2-2013.

In my remarks about sheet 2 I wrote twice $I/R$. That should have been $R/I$.

The next two sections will be more explicit. We will mainly be dealing with $\mathbb Z$, $\mathbb Q[X]$, $\mathbb Z[X]$ and subfields of $\mathbb C$ that are finite dimensional as vector spaces over $\mathbb Q$. - The lecture on Monday 4-3-2013.

In the description of the Euclidean algorithm for the computation of the gcd I said “$R=\mathbb Z$ or $F[X]$”, but then I wrote $\deg(r_k)$ which doesn't make sense for integers. However, the algorithm and Thm 4 work for any Euclidean domain. I won't say what that is exactly, you can find a precise definition in the literature, but in vague terms it is an integral domain which has a degree function and a division with remainder “algorithm” (actually quotient and remainder are not required to be unique). The integers form a Euclidean domain with the absolute value as degree function. So what I wrote makes sense for the integers if you repace $\deg$ by the absolute value. I think, however, that it would be more natural in the case $R=\mathbb Z$ to just omit $\deg$ and take all $r_k\ge0$: $r_{-1}=|a|$, $r_0=|b|$ and $r_{k+1}$ is the remainder under division of $r_{k-1}$ by $r_k$, so $0\le r_{k+1}< r_k$. - The lecture on Tuesday 5-3-2013.

For those of you who still struggle with calculating modulo ideals: For a subgroup $H$ of of a group $G$ be have $xH=yH$ iff $y^{-1}x\in H$. So for an ideal $I$ of a ring $R$ we have $x+I=y+I$ iff $x-y\in I$: we apply the results from the first part of the course with additive notation. Note also that the zero element of $R/I$ is $I$ and the unit element is $1+I$.

For the exercises on sheet 2 you may use that in every PID we have unique factorisation into irreducibles. We will see next lecture that the minimal polynomial over $\mathbb Q$ of a complex number $\alpha$ which is algebraic over $\mathbb Q$ is characterised by the properties that it is monic, irreducible and zero at $\alpha$ (It should now already be clear that these conditions are sufficient). This should be enough to do all the exercises on sheet 2.

A hint for question 1(ii): First show that for a maximal ideal $I$ of $R$ and $a\in R\setminus I$ we have $I+Ra=R$. Extra hints for question 4: First note that $X^4+64$ is either irreducible over $\mathbb Q$ or the product of two irreducible polynomials of degree two in $\mathbb Q[X]$. In the latter case each of these must be a product of two linear (= of degree 1) factors over $\mathbb C$. To factor $X^4+64$ into irreducibles over $\mathbb C$ you could begin by observing that over $\mathbb C$ it is the difference of two squares ($a^2+b^2=a^2-(ib)^2$). - The lecture on Thursday 7-3-2013.

So the two main applications of the fact that $F[X]$ is a PID are- The definition of the gcd (and lcm) in $F[X]$. This leads via Prop 5(1) to the proof that $F[X]$ is a UFD (Thm 7).
- The definition of the minimal polynomial for an element $a$ in an extension ring $R$ of $F$ ($R$ commutative, or, more generally, $F\subseteq Z(R)$).

**Def**. Let $R$ be an integral domain and let $P$ be a set of $\sim$-representatives for the irreducible elements (if $R=F[X]$, then we can take $P$ the set of monic irreducible polynomials). Then $R$ is called a*unique factorisation domain*(UFD) if for every $a\in R$ there exist unique integers $k_p\ge0$, $p\in P$, finitely many nonzero, such that $a\sim\prod_{p\in P}p^{k_p}$. - The lecture on Thursday 14-3-2013.

When I say “....an extension $F\subseteq K$....” the word extension refers to the whole expression $F\subseteq K$,**not**just $F$. So I say: $K$ is an extension of $F$ or $F\subseteq K$ is an extension, where, in the latter case, I refer to the whole expression $F\subseteq K$.

If you find it awkward to work with the ring $F[X]/(f_\alpha)$, then you can identify it with the space of polynomials of degree $<\deg(f_\alpha)$ with obvious addition and with multiplication: $f\cdot g\stackrel{\rm def}{=}$ the remainder of $fg$ under division by $f_\alpha$. Indeed each class modulo $(f_\alpha)$ has a unique representant (element) of degree $<\deg(f_\alpha)$. Note that the underlying vector space only depends on $\deg(f_\alpha)$, but the multiplication depends on $f_\alpha$ itself. - The lecture on Monday 18-3-2013.

Please look again at the definition of “ subring generated by” and at Prop II.6 and its corollary. In fact Prop II.6 is one of the most fundamental results of the course. - The lecture on Thurday 21-3-2013.

I was hesitating a bit when I wanted to show that $X^4+1$ is irreducible over $\mathbb Q$. I had in mind to show that it is irreducible modulo $3$ and then apply Cor. 3 to Prop. 8 (Gauss' lemma). Indeed $X^4+1$ has no roots in $\mathbb F_3$, but since it is of degree $4$ that doesn't prove that it is irreducible over $\mathbb F_3$. In fact it factorises modulo $3$ as $X^4+1=(X^2+X-1)(X^2-X-1)$! Fortunately, the implication in the above corollary is only in one direction. It still follows by Eisenstein's criterion with $p=2$ that $(X+1)^4+1$ is irreducible over $\mathbb Q$. So $X^4+1$ is also irreducible over $\mathbb Q$. - The lecture on Tuesday 26-3-2013.

What the proof of Prop II.6 and even more the proof of Thm III.5 seem to be saying is that the roots of an irreducible polynomial are all “ the same”.

What I showed at the end of the lecture is that for a finite normal extension $F\subseteq K$ and for any finite ${\rm Gal}(K/F)$-stable subset $\mathcal O$ of $K$, the polynomial $\prod_{\alpha\in\mathcal O}(X-\alpha)$ is in $F[X]$.

For example, we can take $K=\mathbb Q(\zeta)$, $\zeta$ a primitive $n$-th root of unity. Then the set $\mathcal O_n=\{\zeta^m\,|\,0\le m < n, {\rm gcd}(m,n)=1\}$ of primitive $n$-th roots of unity is a finite ${\rm Gal}(K/\mathbb Q)$-stable subset of $K$. So $\Phi_n(X)=\prod_{\xi\ \text{of order}\ n}(X-\xi)$ is in $\mathbb Q[X]$. Since it is a monic divisor of $X^n-1$, it follows by Cor 2 to Prop II.8 that it is in $\mathbb Z[X]$. The polynomial $\Phi_n$ is called the*$n$-th cyclotomic polynomial*. One can show that it is irreducible over $\mathbb Q$ (we did this for $n$ a prime). So $\mathcal O_n$ is actually a single ${\rm Gal}(K/\mathbb Q)$-orbit. - The lecture on Thursday 28-3-2013.

For those interested in finite fields: The Galois groups of finite extensions of finite fields are always cyclic.

If $K$ is given as $K=F(\alpha_1,...,\alpha_n)$, then any $F$-linear automorphism is determined by what it does to the $\alpha_i$. Furthermore, an element of $K$ can only be moved to one of its conjugates over $F$. So if $\alpha_i$ has $m_i$ conjugates over $F$ (including itself), then that gives us $m_1\cdots m_n$