Examples in Galois Theory 3 - Polynomials of Prime Degree and Pairs of Nonreal Roots

In this episode we focus on the rational field. What can we know about the Galois group of an irreducible polynomial with prime degree? There is a method by counting the number of nonreal roots. From this, we obtain an algorithm to compute the Galois group.

Examples in Galois Theory 2 - Cubic Extensions

We study the Galois group of a cubic polynomial over a field with characteristic not equal to 2 and 3.

Examples in Galois Theory 1 - Complex Field is Algebraically Closed

We try to prove the fundamental theorem of algebra, that the complex field is algebraically closed, using as little analysis as possible. In other words, the following proof will be *almost* algebraic.

The Grothendienck Group

Free group

Let $A$ be an abelian group. Let $(e_i)_{i \in I}$ be a family of elements of $A$. We say that this family is a basis for $A$ if the family is not empty, and if every element of $A$ has a unique expression as a linear expression $x = \sum_{i \in I} x_i e_i$ where $x_i \in \mathbb{Z}$ and almost all $x_i$ are equal to $0$. This means that the sum is actually finite. An abelian group is said to be free if it has a basis. Alternatively, we may write $A$ as a direct sum by $A \cong \bigoplus_{i \in I}\mathbb{Z}e_i.$

Free abelian group generated by a set

Let $S$ be a set. Say we want to get a group out of this for some reason, so how? It is not a good idea to endow $S$ with a binary operation beforehead since overall $S$ is merely a set. We shall generate a group out of $S$ in the most freely way.

Let $\mathbb{Z}\langle S \rangle$ be the set of all maps $\varphi:S \to \mathbb{Z}$ such that, for only a finite number of $x \in S$, we have $\varphi(x) \neq 0$. For simplicity, we denote $k \cdot x$ to be some $\varphi_0 \in \mathbb{Z}\langle S \rangle$ such that $\varphi_0(x)=k$ but $\varphi_0(y) = 0$ if $y \neq x$. For any $\varphi$, we claim that $\varphi$ has a unique expression $\varphi=k_1 \cdot x_1 + k_2 \cdot x_2 + \cdots + k_n \cdot x_n.$ One can consider these integers $k_i$ as the order of $x_i$, or simply the time that $x_i$ appears (may be negative). For $\varphi\in\mathbb{Z}\langle S \rangle$, let $I=\{x_1,x_2,\cdots,x_n\}$ be the set of elements of $S$ such that $\varphi(x_i) \neq 0$. If we denote $k_i=\varphi(x_i)$, we can show that $\psi=k_1 \cdot x_1 + k_2 \cdot x_2 + \cdots + k_n \cdot x_n$ is equal to $\varphi$. For $x \in I$, we have $\psi(x)=k$ for some $k=k_i\neq 0$ by definition of the '$\cdot$'; if $y \notin I$ however, we then have $\psi(y)=0$. This coincides with $\varphi$. $\blacksquare$

By definition the zero map $\mathcal{O}=0 \cdot x \in \mathbb{Z}\langle S \rangle$ and therefore we may write any $\varphi$ by $\varphi=\sum_{x \in S}k_x\cdot x$ where $k_x \in \mathbb{Z}$ and can be zero. Suppose now we have two expressions, for example $\varphi=\sum_{x \in S}k_x \cdot x=\sum_{x \in S}k_x'\cdot x$ Then $\varphi-\varphi=\mathcal{O}=\sum_{x \in S}(k_x-k'_x)\cdot x$ Suppose $k_y - k_y' \neq 0$ for some $y \in S$, then $\mathcal{O}(y)=k_y-k_y'\neq 0$ which is a contradiction. Therefore the expression is unique. $\blacksquare$

This $\mathbb{Z}\langle S \rangle$ is what we are looking for. It is an additive group (which can be proved immediately) and, what is more important, every element can be expressed as a 'sum' associated with finite number of elements of $S$. We shall write $F_{ab}(S)=\mathbb{Z}\langle S \rangle$, and call it the free abelian group generated by $S$. For elements in $S$, we say they are free generators of $F_{ab}(S)$. If $S$ is a finite set, we say $F_{ab}(S)$ is finitely generated.

An abelian group is free if and only if it is isomorphic to a free abelian group $F_{ab}(S)$ for some set $S$.

Proof. First we shall show that $F_{ab}(S)$ is free. For $x \in M$, we denote $\varphi = 1 \cdot x$ by $[x]$. Then for any $k \in \mathbb{Z}$, we have $k[x]=k \cdot x$ and $k[x]+k'[y] = k\cdot x + k' \cdot y$. By definition of $F_{ab}(S)$, any element $\varphi \in F_{ab}(S)$ has a unique expression $\varphi = k_1 \cdot x_1 + \cdots + k_n \cdot x_n =k_1[x_1]+\cdots+k_n[x_n]$ Therefore $F_{ab}(S)$ is free since we have found the basis $([x])_{x \in S}$.

Conversely, if $A$ is free, then it is immediate that its basis $(e_i)_{i \in I}$ generates $A$. Our statement is therefore proved. $\blacksquare$

The connection between an arbitrary abelian group an a free abelian group

(Proposition 1) If $A$ is an abelian group, then there is a free group $F$ which has a subgroup $H$ such that $A \cong F/H$.

Proof. Let $S$ be any set containing $A$. Then we get a surjective map $\gamma: S \to A$ and a free group $F_{ab}(S)$. We also get a unique homomorphism $\gamma_\ast:F_{ab}(S) \to A$ by \begin{aligned} \gamma_\ast:F_{ab}(S) &\to A \\ \varphi=\sum_{x \in S}k_x\cdot x &\mapsto \sum_{x \in S}k_x\gamma(x) \end{aligned} which is also surjective. By the first isomorphism theorem, if we set $H=\ker(\gamma_\ast)$ and $F_{ab}(S)=F$, then $F/H \cong A.$ $\blacksquare$

(Proposition 2) If $A$ is finitely generated, then $F$ can also be chosen to be finitely generated.

Proof. Let $S$ be the generator of $A$, and $S'$ is a set containing $S$. Note if $S$ is finite, which means $A$ is finitely generated, then $S'$ can also be finite by inserting one or any finite number more of elements. We have a map from $S$ and $S'$ into $F_{ab}(S)$ and $F_{ab}(S')$ respectively by $f_S(x)=1 \cdot x$ and $f_{S'}(x')=1 \cdot x'$. Define $g=f_{S'} \circ \lambda:S' \to F_{ab}(S)$ we get another homomorphism by \begin{aligned} g_\ast:F_{ab}(S') &\to F_{ab}(S) \\ \varphi'=\sum_{x \in S'}k_{x} \cdot x &\mapsto \sum_{x \in S'}k_{x}\cdot g(x) \end{aligned} This defines a unique homomorphism such that $g_\ast \circ f_{S'} = g$. As one can also verify, this map is also surjective. Therefore by the first isomorphism theorem we have $A \cong F_{ab}(S) \cong F_{ab}(S')/\ker(g_\ast)$ $\blacksquare$

It's worth mentioning separately that we have implicitly proved two statements with commutative diagrams:

(Proposition 3 | Universal property) If $g:S \to B$ is a mapping of $S$ into some abelian group $B$, then we can define a unique group-homomorphism making the following diagram commutative:

(Proposition 4) If $\lambda:S \to S$ is a mapping of sets, there is a unique homomorphism $\overline{\lambda}$ making the following diagram commutative:

(In the proof of Proposition 2 we exchanged $S$ an $S'$.)

The Grothendieck group

(The Grothendieck group) Let $M$ be a commutative monoid written additively. We shall prove that there exists a commutative group $K(M)$ with a monoid homomorphism $\gamma:M \to K(M)$

satisfying the following universal property: If $f:M \to A$ is a homomorphism from $M$ into a abelian group $A$, then there exists a unique homomorphism $f_\gamma:K(M) \to A$ such that $f=f_\gamma\circ\gamma$. This can be represented by a commutative diagram:

Proof. There is a commutative diagram describes what we are doing.

Let $F_{ab}(M)$ be the free abelian group generated by $M$. For $x \in M$, we denote $1 \cdot x \in F_{ab}(M)$ by $[x]$. Let $B$ be the group generated by all elements of the type $[x+y]-[x]-[y]$ where $x,y \in M$. This can be considered as a subgroup of $F_{ab}(M)$. We let $K(M)=F_{ab}(M)/B$. Let $i=x \to [x]$ and $\pi$ be the canonical map $\pi:F_{ab}(M) \to F_{ab}(M)/B.$ We are done by defining $\gamma: \pi \circ i$. Then we shall verify that $\gamma$ is our desired homomorphism satisfying the universal property. For $x,y \in M$, we have $\gamma(x+y)=\pi([x+y])$ and $\gamma(x)+\gamma(y) = \pi([x])+\pi([y])=\pi([x]+[y])$. However we have $[x+y]-[x]-[y] \in B,$ which implies that $\gamma(x)+\gamma(y)=\pi([x]+[y])=\pi([x+y]) = \gamma(x+y).$ Hence $\gamma$ is a monoid-homomorphism. Finally the universal property. By proposition 3, we have a unique homomorphism $f_\ast$ such that $f_\ast \circ i = f$. Note if $y \in B$, then $f_\ast(y) =0$. Therefore $B \subset \ker{f_\ast}$ Therefore we are done if we define $f_\gamma(x+B)=f_\ast (x)$. $\blacksquare$

Why such a $B$? Note in general $[x+y]$ is not necessarily equal to $[x]+[y]$ in $F_{ab}(M)$, but we don't want it to be so. So instead we create a new equivalence relation, by factoring a subgroup generated by $[x+y]-[x]-[y]$. Therefore in $K(M)$ we see $[x+y]+B = [x]+[y]+B$, which finally makes $\gamma$ a homomorphism. We use the same strategy to generate the tensor product of two modules later. But at that time we have more than one relation to take care of.

Cancellative monoid

If for all $x,y,z \in M$, $x+y=x+z$ implies $y=z$, then we say $M$ is a cancellative monoid, or the cancellation law holds in $M$. Note for the proof above we didn't use any property of cancellation. However we still have an interesting property for cancellation law.

(Theorem) The cancellation law holds in $M$ if and only if $\gamma$ is injective.

Proof. This proof involves another approach to the Grothendieck group. We consider pairs $(x,y) \in M \times M$ with $x,y \in M$. Define $(x,y) \sim (x',y') \iff \exists \ell \in M, y+x'+\ell=x+y'+\ell.$ Then we get a equivalence relation (try to prove it yourself!). We define the addition component-wise, that is, $(x,y)+(x',y')=(x+x',y+y')$, then the equivalence classes of pairs form a group $A$, where the zero element is $[(0,0)]$. We have a monoid-homomorphism $f:x \mapsto [(x,0)].$ If cancellation law holds in $M$, then \begin{aligned} f(x) = f(y) &\implies [(x,0)] = [(y,0)] \\ &\implies 0+y+\ell=x+0+\ell \\ &\implies x=y. \end{aligned} Hence $f$ is injective. By the universal property of the Grothendieck group, we get a unique homomorphism $f_\gamma$ such that $f_\gamma \circ \gamma = f$. If $x \neq 0$ in $M$, then $f_\gamma \circ \gamma(x) \neq 0$ since $f$ is injective. This implies $\gamma(x) \neq 0$. Hence $\gamma$ is injective.

Conversely, if $\gamma$ is injective, then $i$ is injective (this can be verified by contradiction). Then we see $f=f_\ast \circ i$ is injective. But $f(x)=f(y)$ if and only if $x+\ell = y+\ell$, hence $x+ \ell = y+ \ell$ implies $x=y$, the cancellation law holds on $M$.

Examples

Our first example is $\mathbb{N}$. Elements of $F_{ab}(\mathbb{N})$ are of the form $\varphi=k_1 \cdot n_1 + k_2 \cdot n_2+\cdots + k_m \cdot n_m.$ For elements in $B$ they are generated by $\varphi=1\cdot (m+n)-1\cdot m - 1\cdot n$ which we wish to represent $0$. Indeed, $K(\mathbb{N}) \simeq \mathbb{Z}$ since if we have a homomorphism \begin{aligned} f:K(\mathbb{N}) &\to \mathbb{Z} \\ \sum_{j=1}^{m}k_j \cdot n_j +B &\mapsto \sum_{j=1}^{m}k_j n_j. \end{aligned} For $r \in \mathbb{Z}$, we see $f(1 \cdot r+B)=r$. On the other hand, if $\sum_{j=1}^{m}k_j \cdot n_j \not\in B$, then its image under $f$ is not $0$.

In the first example we 'granted' the natural numbers 'subtraction'. Next we grant the division on multiplicative monoid.

Consider $M=\mathbb{Z} \setminus 0$. Now for $F_{ab}(M)$ we write elements in the form $\varphi={}^{k_1}n_1{}^{k_2}n_2\cdots{}^{k_m}n_m$ which denotes that $\varphi(n_j)=k_j$ and has no other differences. Then for elements in $B$ they are generated by $\varphi = {}^1(n_1n_2){}^{-1}(n_1)^{-1}(n_2)$ which we wish to represent $1$. Then we see $K(M) \simeq \mathbb{Q} \setminus 0$ if we take the isomorphism \begin{aligned} f:K(M) &\to \mathbb{Q} \setminus 0 \\ \left(\prod_{j=1}^{m}{}^{k_j}n_j\right)B &\mapsto \prod_{j=1}^{m}n_j^{k_j}. \end{aligned}

Of course this is not the end of the Grothendieck group. But for further example we may need a lot of topology background. For example, we have the topological $K$-theory group of a topological space to be the Grothendieck group of isomorphism classes of topological vector bundles. But I think it is not a good idea to post these examples at this timing.

Cauchy sequence in group theory

Recall - Cauchy sequence in analysis

Before we go into group theory, let's recall how Cauchy sequence is defined in analysis.

Real/complex number

A sequence $(x_n)_{n=1}^{\infty}$ of real/complex numbers is called a Cauchy sequence if, for every $\varepsilon>0$, there is a positive integer $N$ such that for all $m,n>N$, we have $|x_m-x_n|<\varepsilon.$ That said, the distance between two numbers is always 'too close'. Notice that only distance is involved, the definition of Cauchy sequence in metric space comes up in the natural of things.

Metric space

Given a metric space $(X,d)$, a sequence $(x_n)_{n=1}^{\infty}$ is Cauchy if for every real number $\varepsilon>0$, there is a positive integer $N$ such that, for all $m,n>N$, the distance by $d(x_m,x_n)<\varepsilon.$

Topological vector space

By considering the topology induced by metric, we see that $x_n$ lies in a neighborhood of $x_m$ with radius $\varepsilon$. But a topology can be constructed by neighborhood, hence the Cauchy sequence for topological vector space follows.

For a topological vector space $X$, pick a local base $\mathcal{B}$, then $(x_n)_{n=1}^{\infty}$ is a Cauchy sequence if for each member $U \in \mathcal{B}$, there exists some number $N$ such that for $m,n>N$, we have $x_m-x_n \in U.$ But in a topological space, it's not working. Consider two topological space by $X=(0,1)\quad Y=(1,+\infty)$ with usual topology. We have $X \simeq Y$ since we have the map by \begin{aligned} h:X &\to Y \\ x &\mapsto \frac{1}{x} \end{aligned} as a homeomorphism. Consider the Cauchy sequence $(\frac{1}{n+1})_{n=1}^{\infty}$, we see $(h(\frac{1}{n+1}))_{n=1}^{\infty}=(n+1)_{n=1}^{\infty}$ which is not Cauchy. This counterexample shows that being a Cauchy sequence is not preserved by homeomorphism.

Topological group

Similarly, one can have a Cauchy sequence in a topological group (bu considering subtraction as inverse).

A sequence $(x_n)_{n=1}^{\infty}$ in a topological group $G$ is a Cauchy sequence if for every open neighborhood $U$ of the identity $G$, there exists some number $N$ such that whenever $m,n>N$, we have $x_nx_m^{-1} \in U$

Completeness

A metric space $(X,d)$ where every Cauchy sequence converges is complete.

Spaces like $\mathbb{R}$, $\mathbb{C}$ are complete with Euclid metric. But consider the sequence in $\mathbb{Q}$ by $a_n=\left(1+\frac{1}{n}\right)^{n}$ we have $a_n\in\mathbb{Q}$ for all $n$ but the sequence does not converge in $\mathbb{Q}$. Indeed in $\mathbb{R}$ we can naturally write $a_n \to e$ but $e \notin \mathbb{Q}$ as we all know.

There are several ways to construct $\mathbb{R}$ from $\mathbb{Q}$. One of the most famous methods is Dedekind's cut. However you can find no explicit usage of Cauchy sequence. There is another method by using Cauchy sequence explicitly. We are following that way algebraically.

Completing a group

Suppose we are given a group $G$ with a sequence of normal subgroups $(H_n)_{n=1}^{\infty}$ with $H_n \supset H_{n+1}$ for all $n$, all of which has finite index. We are going to complete this group.

Cauchy sequence

A sequence $(x_n)_{n=1}^{\infty}$ in $G$ will be called Cauchy sequence if given $H_k$, there exists some $N>0$ such that for $m,n>N$, we have $x_nx_m^{-1} \in H_k$ Indeed, this looks very similar to what we see in topological group, but we don't want to grant a topology to the group anyway. This definition does not go to far from the original definition of Cauchy sequence in $\mathbb{R}$ as well. If you treat $H_k$ as some 'small' thing, it shows that $x_m$ and $x_n$ are close enough (by considering $x_nx_m^{-1}$ as their difference).

Null sequence

A sequence $(x_n)_{n=1}^{\infty}$ in $G$ will be called null sequence if given $k$, there exists some $N>0$ such that for all $n>N$, we have $x_n\in H_k$ or you may write $x_ne^{-1} \in H_k$. It can be considered as being arbitrarily close to the identity $e$.

Cauchy sequence and null sequence as groups

The Cauchy sequences (of $G$) form a group under termwise product

Proof. Let $C$ be the set of Cauchy sequences, we shall show that $C$ forms a group. For $(x_1,x_2,\cdots),(y_1,y_2,\cdots)\in C$, the product is defined by $(x_1,x_2,\cdots)(y_1,y_2,\cdots)=(x_1y_1,x_2y_2,\cdots)$ The associativity follows naturally from the associativity of $G$. To show that $(x_1y_1,x_2y_2,\cdots)$ is still a Cauchy sequence, notice that for big enough $m$, $n$ and some $k$, we have $x_nx_m^{-1}\in H_k \quad y_ny_m^{-1}\in H_k.$ But $(x_ny_n)(x_my_m)^{-1}=x_ny_ny_m^{-1}x_m^{-1}$. To show that this is an element of $H_k$, notice that $x_ny_ny_m^{-1}x_m^{-1}=(x_ny_ny_m^{-1}{\color{red}{x_n^{-1}}})({\color{red}{x_n}}x_m^{-1})$ Since $y_ny_m^{-1}\in H_k$, $H_k$ is normal, we have $x_ny_ny_mx_n^{-1} \in H_k$. Since $x_nx_m^{-1} \in H_k$, $(x_ny_n)(x_my_m)^{-1}$ can be viewed as a product of two elements of $H_k$, therefore is an element of $H_k$.

Obviously, if we define $e_C=(e_G,e_G,\cdots)$, where $e_G$ is the identity of $G$, $e_C$ becomes the identity of $C$, since $(x_1,x_2,\cdots)(e_G,e_G,\cdots)=(x_1,x_2,\cdots)=(e_G,e_G,\cdots)(x_1,x_2,\cdots).$ Finally the inverse. We need to show that $(x_1,x_2,\cdots)^{-1}=(x_1^{-1},x_2^{-1},\cdots)$ is still an element of $C$. This is trivial since if we have $x_nx_m^{-1} \in H_k$ then $(x_n^{-1})(x_m^{-1})^{-1}=x_n^{-1}x_m \in H_k$ as $H_k$ is a group.

The null sequences (of $G$) form a group, further, it's a normal subgroup of $C$, that is, the group of Cauchy sequences.

Let $N$ be the set of null sequences of $G$. Still, the identity is defined by $(e_G,e_G,\cdots)$, and there is no need to duplicate the validation. And the associativity still follows from $G$. To show that $N$ is closed under termwise product, namely if $(x_n),(y_n) \in N$, then $(x_ny_n)\in N$, one only need to notice that, for big $n$, we already have $x_n,y_n \in H_k.$ Therefore $x_ny_n \in H_k$ since $x_n$ and $y_n$ are two elements of $H_k$.

To show that $(x_n^{-1})$, which should be treated as the inverse of $(x_n)$, is still in $N$, notice that if $x_n \in H_k$, then $x_n^{-1} \in H_k$.

Next, we shall show that $N$ is a subgroup of $C$, which is equivalent to show that every null sequence is Cauchy. Given $H_p \supset H_q$, for $(x_n)\in{N}$, there are some big enough $m$ and $n$ such that $x_n \in H_p \quad x_m \in H_q$ therefore $x_nx_m^{-1} \in H_p$ as desired. Finally, pick $(p_n) \in N$ and $(q_n) \in C$, we shall show that $(q_n)(p_n)(q_n)^{-1} \in N$. That is, the sequence $(q_np_nq_n^{-1})$ is a null sequence. Given $H_k$, we have some big $n$ such that $p_n \in H_k$ therefore $q_np_nq_n^{-1} \in H_k$ since $H_k$ is normal. Our statement is proved.

The factor group

The factor group $C/N$ is called the completion of $G$ (with respect to $(H_n)$).

As we know, the elements of $C/N$ are cosets. A coset can be considered as an element of $G$'s completion. Let's head back to some properties of factor group. Pick $x,y \in C$, then $xN=yN$ if and only if $x^{-1}y \in N$. With that being said, two Cauchy sequences are equivalent if their 'difference' is a null sequence.

Informally, consider the addictive group $\mathbb{Q}$. There are two Cauchy sequence by $(1,1,1,\cdots,1,\cdots) \\ (0.9,0.99,0.999,0.9999,\cdots).$ They are equivalent since $(0.1,0.01,0.001,\cdots)$ is a null sequence. That's why people say $0.99999... = 1$ (in analysis, the difference is convergent to $0$; but in algebra, we say the two sequences are equivalent). Another example, $\ln{2}$ can be represented by the equivalent class of $(0.6,0.69,0.693,\cdots).$

Remarks

We made our completion using Cauchy sequences. The completion is filled with some Cauchy sequence and some additions of 'nothing', whence the gap disappears.

Again, the sequence of normal subgroups does not have to be indexed by $\mathbb{N}$. It can be indexed by any directed partially ordered set, or simply partially ordered set. Removing the restriction of index set gives us a great variety of implementation.

However, can we finished everything about completing $\mathbb{Q}$ using this? The answer is, no - the multiplication is not verified! To finish this, field theory have to be taken into consideration.