TITLE: Showing that the Hopf fibration is a non-trivial fibre bundle
QUESTION [6 upvotes]: I want to show that the Hopf bundle
$$ \mathbb{S}^1 \rightarrow \mathbb{S^3} \rightarrow \mathbb{S}^2$$
is non-trivial as a principal fibre bundle.
I have seen hints of several different approaches:

Hopfs original approach, using linking numbers, see Hopf fibration and $\pi_3(\mathbb{S}^2)$. 
Hopf invariant.
Cohomology. 

I want to keep it simple. My gut says cohomology is my best bet (I am slightly familiar with de Rahm Cohomology). Unfortunately, I am having trouble finding sources that treat this on my level (without a lot of general theory I think I don't need).  
I have the following books:

The Topology of Fibre Bundles, Steenrod.
Fibre Bundles, Husemoller.
Manifold and Differential Geometry, Jeffry Lee.

What is the bare minimum of theory I need to to get to this result? What route would you advise?
I am an undergraduate working on a Bsc thesis.

REPLY [11 votes]: The easier way to show that is to remark that $\pi_1(S_3)=1$ and $\pi_1(S_2\times S_1)=Z$.