Characters in Analysis and Algebra

Let $M$ be a monoid and $K$ be a field, then by a character of $G$ in $K$ we mean a monoid-homomorphism

$\chi:M \to K^\ast.$ By trivial character we mean a character such that $\chi(M)=\{1\}$. We are particularly interested in the linear independence of characters. Functions $f_i:M \to K$ are called linearly independent over $K$ if whenever $a_1f_1+\cdots+a_nf_n=0$ with all $a_i \in K$, we have $a_i=0$ for all $i$. $\def\Tr\operatorname{Tr}$

Character in Fourier Analysis

In Fourier analysis we are always interested by functions like $f(x)=e^{-inx}$ or $g(x)= e^{-ixt}$, corresponding to Fourier series (integration on $\mathbb{R}/2\pi\mathbb{Z}$) and Fourier transform. Later mathematicians realised that everything can be set in a locally compact abelian (LCA) group. For this reason we need to generalise these functions, and the bounded ones coincide with our definition of characters.

Let $G$ be a LCA group, then $\gamma:G \to \mathbb{C}$ is called a character if $|\gamma(x)|=1$ for all $x \in G$ and $\gamma(x+y)=\gamma(x)\gamma(y).$ Note since $G$ is automatically a monoid, this coincide with our ordinary definition of character. The set of continuous characters form a group $\Gamma$, which is called the dual group of $G$.

If $G=\mathbb{R}$, solving the equation $\gamma(x+y)=\gamma(x)\gamma(y)$ in whatever way he or she likes we obtain $\gamma(x)=e^{Ax}$ for some $A \in \mathbb{C}$. But $|e^{Ax}| \equiv 1$ (or merely being bounded) forces $A$ to be purely imaginary, say $A=it$, then we have $\gamma(x)=e^{itx}$. Hence the dual group of $\mathbb{R}$ can be determined by (the speed of) rotation on the unit circle.

With this we have our generalised version of Fourier transform. Let $G$ be a LCA group, $f \in L^1(G)$, then the Fourier transform is given by $\hat{f}(\gamma) = \int_G f(x)\gamma(-x)dx, \quad \gamma \in \Gamma.$ One can intuitively verify that $\hat{f}$ is exactly the Gelfand transform of $f$, the step of which will be sketched below. On one hand, one can indeed verify that $f \mapsto \hat{f}(\gamma)$ is indeed a Banach algebra homomorphism $L^1(G) \to \mathbb{C}$, for all $\gamma \in \Gamma$. This is a plain application of Fubini's theorem. On the other hand, let $h:L^1(G) \to \mathbb{C}$ be any non-trivial Banach algebra homomorphism. One can investigate that $\| h \| =1$ and hence $h$ is a bounded linear functional. By Riesz's representation theorem, there is some $\phi \in L^\infty(G)$ with $\| \phi\|_\infty = 1$ such that $h(f) = \int_G f(x)\phi(x)dx.$ We can indeed assume that $\phi$ is continuous. With $h$ being algebra homomorphism, we can see $\phi(x+y)=\phi(x)\phi(y).$ We know that $|\phi(x)| \le 1$ but $\phi(-x)=\phi(x)^{-1}$ forces $|\phi(x)|=1$. The proof is done after some routine verification of uniqueness.

Indeed, with this identification, we can also identify $\Gamma$ as the maximal ideal space of $L^1(G)$, which results in the following interesting characterisation.

If $G$ is discrete, then $\Gamma$ is compact; if $G$ is compact, then $\Gamma$ is discrete.

Proof. If $G$ is discrete, then $L^1(G)$ has a unit. The maximal ideal space, which can be identified as $\Gamma$, is a compact Hausdorff space.

If $G$ is compact, then its Haar measure can be normalised so that $m(G)=1$. We prove that the singleton containing the unit alone is an open set. Let $\gamma \in \Gamma$ be a character $\ne 1$, then there exists some $x_0$ such that $\gamma(x_0) \ne 1$. As a result, $\int_G \gamma(x)dx = \gamma(x_0)\int_G \gamma(x-x_0)dx = \gamma(x_0)\int_G \gamma(x)dx$ and hence $\int_G\gamma(x)dx=0$. If $\gamma=1$ then $\int_G \gamma(x)=1$.

Besides, the compactness of $G$ implies the constant function $f \equiv 1$ is in $L^1(G)$. As a result, $\hat{f}(1)=1$ but $\hat{f}(\gamma)=0$ whenever $\gamma \ne 1$. But $\hat{f}$ is continuous, $\{\gamma:{f}(\gamma) \ne 0\}=\{1\}$ is open. $\square$

Linear Independence of Characters

If characters of $G$ are linear independent, then they are pairwise distinct, but what about the converse? Dedekind answered this question affirmatively. But his approach is rather complicated: it needed determinant. However, Artin found a neat way to do it:

Theorem (Dedekind-Artin) Let $M$ be a monoid and $K$ a field. Let $\chi_1,\dots,\chi_n$ be distinct characters of $G$ in $K$. Then they are linearly independent over $K$.

Proof. Suppose this is false. Let $N$ be the smallest integer that $a_1\chi_1+a_2\chi_2+\cdots+a_N\chi_N = 0$ but not all $a_i$ are $0$, for distinct $\chi_i$. Since $\chi_1 \ne \chi_2$, there is some $z \in M$ such that $\chi_1(z) \ne \chi_2(z)$. Yet still we have $a_1\chi_1(zx)+\cdots+a_N\chi_N(zx)=0.$ Since $\chi_i$ are characters, for all $x \in M$ we have $a_1\chi_1(z)\chi_1(x)+\cdots+a_N\chi_N(z)\chi_N(x)=0.$ We now have a linear system $\begin{pmatrix} a_1 & a_2 & \cdots & a_N \\ a_1\chi_1(z) & a_2\chi_2(z) & \cdots & a_N\chi_N(z) \end{pmatrix} \begin{pmatrix} \chi_1 \\ \chi_2 \\ \vdots \\ \chi_N \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$ If we perform Gaussian elimination once, we see $\begin{pmatrix} a_1 & a_2 & \cdots & a_N \\ 0 & \left(\frac{\chi_2(z)}{\chi_1(z)}-1\right)a_2 & \cdots & \left(\frac{\chi_N(z)}{\chi_1(z)}-1\right)a_N\chi_N(z) \end{pmatrix} \begin{pmatrix} \chi_1 \\ \chi_2 \\ \vdots \\ \chi_N \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$ But this is to say $\left(\frac{\chi_2(z)}{\chi_1(z)}-1\right)a_2\chi_2 + \cdots + \left(\frac{\chi_N(z)}{\chi_1(z)}-1\right)a_N\chi_N(z)\chi_N=0$ Note by assumption $\frac{\chi_2(z)}{\chi_1(z)}-1 \ne 0$ and therefore we found $N-1$ distinct and linearly independent characters, contradicting our assumption. $\square$

As an application, we consider an $n$-variable equation:

Let $\alpha_1,\cdots,\alpha_n$ be distinct non-zero elements of a field $K$. If $a_1,\cdots,a_n$ are elements of $K$ such that for all integers $v \ge 0$ we have $a_1\alpha_1^v + \cdots + a_n\alpha_n^v = 0$ then $a_i=0$ for all $i$.

Proof. Consider $n$ distinct characters $\chi_i(v)=\alpha^v$ of $\mathbb{Z}_{\ge 0}$ into $K^\ast$. $\square$

Hilbert's Theorem 90

The linear independence of characters gives us a good chance of studying the relation of the field extension and the Galois group.

Hilbert's Theorem 90 (Modern Version) Let $K/k$ be a Galois extension with Galois group $G$, then $H^1(G,K^\ast)=1$ and $H^1(G,K)=0$. This is to say, the first cohomology group is trivial for both addition and multiplication.

It may look confusing but the classic version is about cyclic extensions ($K/k$ is cyclic if it is Galois and the Galois group is cyclic).

Hilbert's Theorem 90 (Classic Version, Multiplicative Form) Let $K/k$ be cyclic of degree $n$ with Galois group $G$ generated by $\sigma$. Then $\frac{\ker N}{1/\sigma{A}} \cong 1$ where $1/\sigma{A}$ consists of all elements of the form $\alpha/\sigma(\alpha)$ with $\alpha \in A$, and $N(\beta)$ is the norm of $\beta \in K$ over $k$.

This corresponds to the statement that $H^1(G,K^\ast)=1$. On the other hand,

Hilbert's Theorem 90 (Classic Version, Additive Form) Let $K/k$ be cyclic of degree $n$ with Galois group $G$ generated by $\sigma$. Then $\frac{\ker \Tr}{(1-\sigma){A}} \cong 0$ where $(1-\sigma)A$ consists of all elements of the form $(1-\sigma)(\alpha)$ with $\alpha \in A$, and $\Tr(\beta)$ is the norm of $\beta \in K$ over $k$.

This corresponds to, of course, the statement that $H^1(G,K)=0$. Note this indeed asserts an exact sequence $0 \to k \to K \xrightarrow{1-\sigma} K \xrightarrow{\Tr} K \to 0.$ Before we prove it we recall what is group cohomology. Let $G$ be a group. We consider the category $G$-mod of left $G$-modules. The set of morphisms of two objects $A$ and $B$, for which we write $\operatorname{Hom}_G(A,B)$, consists of all objects of $G$-set maps from $A$ to $B$. The cohomology groups of $G$ with coefficients in $A$ is the right derived functor of $\operatorname{Hom}_G(\mathbb{Z},-)$: $H^\ast (G,A) \cong \operatorname{Ext}^\ast_{\mathbb{Z}[G]}(\mathbb{Z},A).$ It follows that $H^0(G,A) _G(Z,A)=A/ga-a:g G,a A$. In particular, if $G$ is trivial, then $\operatorname{Hom}_G(\mathbb{Z},-)$ is exact and therefore $H^\ast(G,A)=0$ whenever $\ast \ne 0$. We will see what will happen when $G$ is a Galois group of a Galois extension. If the modern version is beyond your reach, you can refer to the classic version. As a side note, the modern version can also be done using Shapiro's lemma.

Proof of the Modern Version

Proof. Note $\alpha:G \to K^\ast$ is an 1-cocyle if and only if ${}=(_)$ for all $\sigma,\tau \in G$. By Artin's lemma, for each 1-cocyle $\alpha$, the following map is nontrivial: $\Lambda=\sum_{\sigma \in G}\alpha_\sigma \sigma:K \to K.$ Suppose $\gamma=\Lambda(\theta) \ne 0$. Then \begin{aligned} \tau\gamma &= \tau \sum_{\sigma \in G}\alpha_\sigma \sigma(\theta) = \sum_{\sigma \in G}\tau(\alpha_\sigma)\tau\sigma(\theta) = \sum_{\sigma \in G}\alpha_\tau^{-1}\alpha_{\sigma\tau}\tau\sigma(\theta) \\ &= \alpha_\tau^{-1}\sum_{\sigma \in G}\alpha_{\sigma\tau}\sigma\tau(\theta) = \alpha_\tau^{-1}\gamma \end{aligned} which is to say $\alpha_\tau = \gamma/\tau\gamma$. Replacing $\gamma$ with $\gamma^{-1}$ gives what we want: cocycle coincides with coboundary. So much for the multiplicative form.

For the additive form, take $\theta \in K \setminus \ker Tr$. Given a $1$-cocycle $\alpha$ in the additive group $K$, we put $\beta = \frac{1}{\Tr(\theta)}\sum_{\tau \in G}\alpha_\tau \tau(\theta)$ Since cocycle satisfies $\alpha_{\sigma\tau}=\alpha_\sigma+\sigma\alpha_\tau$, we get $\sigma\beta = \frac{1}{\Tr(\theta)}\sum_{\tau \in G}(\alpha_{\sigma\tau}-\alpha_\sigma)\sigma\tau(\theta) = \beta -\alpha_\sigma$ which gives $\alpha_\sigma = \beta-\sigma\beta$. Replacing $\beta$ with $-\beta$ gives what we want. $\square$

Proof of the Classic Version

Additive form. Pick any $\beta-\sigma\beta$, we see $\Tr(\beta-\sigma\beta)=\sum_{\tau \in G}\tau\beta-\sum_{\tau \in G}\tau\beta=0$.

Conversely, assume $\Tr(\alpha)=0$. By Artin's lemma, the trace function is not trivial, hence there exists some $\theta \in K$ such that $\Tr(\theta)\ne 0$, then we take $\beta = \frac{1}{\Tr(\theta)}[\alpha\theta^\sigma+(\alpha+\sigma\alpha)\theta^{\sigma^2}+\cdots+(\alpha+\sigma\alpha+\cdots+\sigma^{n-2}\alpha)\theta^{\sigma^{n-1}}]$ where for convenience we write $\sigma\theta=\theta^\sigma$. Therefore $\beta-\sigma\beta = \frac{1}{\Tr(\theta)}\alpha(\theta+\theta^{\sigma}+\theta^{\sigma^2}+\cdots+\theta^{\sigma^{n-1}})=\alpha$ because other terms are cancelled. $\square$

Multiplicative form. This can be done in a quite similar setting. For any $\alpha=\beta/\sigma\beta$, we have $N(\alpha)=N(\beta)/N(\sigma\beta)=\left(\prod_{\tau \in G}\tau\beta\right)/ \left( \prod_{\tau \in G}\tau\sigma\beta\right)=1.$ Conversely, assume $N(\alpha)=1$. By Artin's lemma, following function is not trivial: $\Lambda:\operatorname{id}+\alpha\sigma+\alpha^{1+\sigma}\sigma^2+\cdots+\alpha^{1+\sigma+\cdots+\sigma^{n-2}}\sigma^{n-1}.$ Suppose now $\beta=\Lambda(\theta) \ne 0$. It follows that \begin{aligned} \alpha\beta^\sigma &= \alpha(\theta+\alpha\theta^\sigma+\cdots+\alpha^{1+\sigma+\cdots+\sigma^{n-2}}\theta^{\sigma^{n-1}})^\sigma \\ &= \alpha(\theta^\sigma+\alpha^\sigma\theta^{\sigma^2}+\cdots+\underbrace{\alpha^{\sigma+\sigma^2+\cdots+\sigma^{n-1}}\theta^{\sigma^n}}_{=\alpha^{-1}\theta}) \\ &= \alpha\theta^\sigma+\alpha^{1+\sigma}+\cdots+\alpha^{1+\sigma+\cdots+\sigma^{n-2}}\theta^{n-1}+\theta \\ &=\beta \end{aligned} and this is exactly what we want. $\square$

Applications

Consider the extension $\mathbb{Q}(i)/\mathbb{Q}$. The Galois group $G=\{1,\tau\}$ is cyclic and generated by $\tau$ the complex conjugation. Now we pick whatever $N(a+bi)=a^2+b^2=1$ where $a,b \in \mathbb{Q}$, we have some $r=s+ti \in \mathbb{Q}(i)$ such that $a+bi = \frac{s+ti}{s-ti}=\frac{s^2-t^2+2sti}{s^2+t^2}= \frac{s^2-t^2}{s^2+t^2}+\frac{2st}{s^2+t^2}i$ If we put $(x,y,z)=(s^2-t^2,2st,s^2+t^2)$, we actually get a Pythagorean triple (if $s,t$ are fractions, we can multiply them with the $\gcd$ of the denominators so they are integers.). Conversely, all Pythagorean triple $(x,y,z)$, we assign it with $\frac{x}{z}+\frac{y}{z}i \in \mathbb{Q}(i)$ then we have an element of norm $1$. Through this we have found all solutions to $x^2+y^2=z^2$. i.e.

Theorem Integers $x,y,z$ satisfy the Diophantine equation $x^2+y^2=z^2$ if and only if $(x,y,z)$ is proportional to $(m^2-n^2,2mn,m^2+n^2)$ for some integers $m,n$.

This can be generalised to all Diophantine equations of the form $x^2+Axy+By^2=Cz^2$ for some nonzero constant $C$ and constant $A,B$ such that the discriminant $A^2-4B$ is square-free. You can find some discussion here.

The additive form is a good friend of "character $p$" things. Artin-Schreier's theorem is a good example of $p$-to-the-$p$.

Theorem (Artin-Schreier) Let $k$ be a field of character $p$ and $K/k$ an extension of degree $p$. Then there exists $\alpha \in K$ and $\alpha$ is the zeroof an equation $X^p-X-a=0$ for some $a \in k$.

Proof. Note the Galois group $G$ of $K/k$ is cyclic and $\Tr(-1)=p(-1)=0$, we are able to use the additive form. Let $\sigma$ be the generator of $G$, there exists some $\alpha \in K$ such that $\sigma\alpha = \alpha+1.$ Hence $\sigma(\sigma(\alpha))=\sigma(\alpha+1)=\alpha+1+1$, and by induction we get $\sigma^i(\alpha) = \alpha+i, \quad i=1,2,\cdots,p$ and $\alpha$ has $p$ conjugates. Therefore $[k(\alpha):k] \ge p$. But in the meantime $[K:k]=[K:k(\alpha)][k(\alpha):k]$ we can only have $[K:k(\alpha)]=1$, which is to say $K=k(\alpha)$. In the meantime, $\sigma(\alpha^p-\alpha)=(\alpha+1)^p-(\alpha+1)=\alpha^p+1^p-\alpha-1 = \alpha^p-\alpha.$ Hence $\alpha^p - \alpha$ lies in the fixed field of $\sigma$, which happens to be $k$. Putting $a=\alpha^p-\alpha$ and our proof is done. $\square$.

For the case when the character is $0$ please see here. There is a converse, which deserves a standalone blog post. It says that the polynomial $f(X)=X^p-X-a$ either has one root in $k$, in which case all its roots are in $k$; or it is irredcible, in which case if $\alpha$ is a root then $k(\alpha)$ is cyclic of degree $p$ over $k$. But I don't know if many people are fans of "character $p$" things.

References

1. Serge Lang, Algbra, Revised Third Edition.
2. Charles A. Weibel, An Introduction to Homological Algebra.
3. Noam D. Elkies, Pythagorean triples and Hilbert’s Theorem 90. (https://abel.math.harvard.edu/~elkies/Misc/hilbert.pdf)
5. Walter Rudin, Fourier Analysis on Groups.

Desvl

2021-10-10

2021-10-20