# Why is number of real characters mod $q$ a multiplicative function?

Let $$R(q)$$ be the number of real characters mod $$q$$. A character $$\chi \mod q$$ is called real if $$\chi(a)\in\mathbb{R}$$ for every $$a\in \mathbb{Z}$$, which means $$\chi(a)\in\{-1,1\}$$ for every $$a\in\mathbb{Z}$$ with gcd$$(a,q)=1$$.

I want to show that this $$R$$ is multiplicative, so $$R(ab)=R(a)R(b)$$ for $$a,b\in\mathbb{Z}$$ with gcd$$(a,b)=1$$. I'm trying to prove this with induced characters, but I'm not getting it completely. Can someone help me to understand it? Thanks!

• Do you mean number of irreducible real characters? Also, how would induced characters help? – A. Pongrácz Dec 30 '18 at 20:37
• @A.Pongrácz not specifically irreducible.. Also, I thought, characters mod $ab$ might be induced by characters mod $a$ and mod $b$? Not sure if this could help – jbuser430 Dec 30 '18 at 20:43
• There are infinitely many real characters $\pmod q$ for every $q$. E.g., any positive integer multiple of the trivial character is a real character. (Check out the definition of a character again.) I believe you want to talk about irreducible characters. – A. Pongrácz Dec 30 '18 at 20:45
• @A.Pongrácz I think we only consider irreducible characters then in my course! :) – jbuser430 Dec 30 '18 at 20:47
• It seems to me that you are also using the word "induced character" in a loose sense. Mind that it has a meaning. How would you "induce" a(n irreducible!) character of $ab$ from a(n irreducible!) character of $a$ and one of $b$? – A. Pongrácz Dec 30 '18 at 20:47