TITLE: Why coefficients in $\mathbb{Q}$ and not $\mathbb{R}$, $\mathbb{C}$, etc.?
QUESTION [5 upvotes]: Hatcher writes in his book "Homology calculations are often simplified by taking coefficients in a field, usually $\mathbb{Q}$ or $\mathbb{Z}_p$ for $p$ prime". 
I wonder why one only consideres $\mathbb{Q}$ as field of char. $0$. I have seen  proofs where one takes coefficients in $\mathbb{Q}$, but I have seen none where one takes coefficients in $\mathbb{R}$ or $\mathbb{C}$. Is there a reason for this? Does coefficients in $\mathbb{Q}$ give more information than coefficients in other fields of char. $0$? Or is it simply a convention to take $\mathbb{Q}$-coefficients?

REPLY [5 votes]: If a cycle $\alpha$ represents a $p$-torsion class so that $\alpha+\ldots\alpha$ ($p$ times) represents the zero class, then working with $\mathbb Q$ coefficients one can form the cycle $\beta=\frac{1}{p}\alpha$ so that $\beta+\ldots+\beta$ ($p$ times) will also necessarily represent the zero class. But $\alpha=p\beta$ and therefore $[\alpha]=0$. Thus you can't have torsion when the coefficients are in $\mathbb Q$. The same goes for the other fields you mentioned but you don't gain any information, though in de Rham theory you have to work with real coefficients, as pointed out in the comments.