I am a beginner in studying the representation theory and I am doing some exercises in this field. So this is not a homework.
My question is about constructing all irreducible representations of quaternion group over the field of rational numbers. I know that this group has four $\mathbb{C}$-irreducible representations of degree 1 (with value in {1,-1}) and one $\mathbb{C}$-irreducible representation of degree 2. It is obvious that its $\mathbb{C}$-irreducible representations of degree 1 are also $\mathbb{Q}$-irreducible representations. Also, we can construct a representation for this group over $\mathbb{Q}$ as follows:
My questions are:
1-What is the best way to show that this representation is irreducible?
2-How can I prove that the only irreducible representations of $Q_8$ over rational field are of degrees 1,1,1,1,4?
Because this is an exercise of section 8.3 of the book "A Course in the Theory of Groups by Robinson", I think that this problem can be solvable using the material contained in it. If I should know something special about representations over $\mathbb{Q}$ and have much information on this topic, please guide me about that.
I with be so appreciate for your helpful answers.