Understanding the Role of Quaternions: Algebra, Normed Linear Space, and Field

[precondition]

I read in some text the following:
Quaternion algebra becomes a normed vector(linear) space with appropriate norm ...(blah blah)... Also since every element has a multiplicative inverse it is a field.

Now, what I find confusing is that according to above a mathematical object called quaternion is not only an algebra but with the norm normed linear space and furthermore a field?? In other words, all these notions of algebra+nvs+field etc should be regarded as some kind of characterisation of an object of our interest in this case quaternion? or further rephrasing this, different ways of looking at an object quaternion?

I would appreciate your comment

 

[D H]

Quaternion multiplication is not commutative. The quaternions form a division ring, not a field.

 

[Hurkyl]

Ah, I think this has what you want to know. (and maybe the results at the bottom of this)