Definition talk:Field Norm of Complex Number

We have Complex Modulus is Norm which suggests that the "norm" is $\sqrt {a^2 + b^2}$. Can this be reviewed? --prime mover (talk) 03:41, 30 July 2017 (EDT)

This is another type of norm, so this definition is unrelated to that theorem. It says "Modulus is Norm" because what is usually called "absolute value" is called "Norm" on $\mathsf{Pr} \infty \mathsf{fWiki}$. Norm has a lot of different meanings. Modulus and Absolute Value do not. --barto (talk) 04:16, 30 July 2017 (EDT)

For example, a "Norm" on the complex numbers could either be an absolute value or a norm when $\C$ is viewed as a vector space over, say, $\R$. Those interpretations are not equivalent. --barto (talk) 04:19, 30 July 2017 (EDT)

In which case it is incorrect to call it "the norm", yeah? Should be "a norm", surely? --prime mover (talk) 05:14, 30 July 2017 (EDT)

Ah, actually it is correct. This norm is neither of those :) It's used in algebraic number theory, mostly. Especially in the introductary parts, before defining the more general field norm. Similarly, a norm is often frist defined for specific quadratic extensions, or for e.g. eisenstein integer (as in Ireland & Rosen, Chapter 9 on reciprocity, and many other books on number theory). Of all norms, field norm is the closest to these. --barto (talk) 06:04, 30 July 2017 (EDT)

Continuing approach

I have found a definition of "Norm" in the book I'm currently studying, defining this as "the norm". I am still not sure about calling this the norm as there are a number.

Maybe we need to add a comment section pointing out the difficulties of this approach. --prime mover (talk) 18:16, 3 April 2019 (EDT)

Field Norm of Complex Number is not a norm according to any definition on the page Definition:Norm because it does not satisfy the (N3) (Triangle Inequality). All indications are that Field Norm of Complex Number is a Field Norm although I do not have a proof of this (and Field Norm of Complex Number Equals Field Norm doesn’t have one either). —Leigh.Samphier (talk) 07:08, 4 April 2019 (EDT)

Okay, so we might do well to rename this page "Field Norm of Complex Number". I confess I don't like this area of mathematics much. --prime mover (talk) 07:29, 4 April 2019 (EDT)

I think you are right, but before committing to this it would be good to have a proof of Field Norm of Complex Number Equals Field Norm. I’ll try to source a proof. Then $\mathsf{Pr} \infty \mathsf{fWiki}$ can make a clear distinction between Field Norms and Norms. At this point in time I don’t see much confusion between “Field Norm” and “Norm on a Field” on $\mathsf{Pr} \infty \mathsf{fWiki}$. It might become a problem in the future.

Field Norms is not an area I have any background in or passion for. -—Leigh.Samphier (talk) 07:51, 4 April 2019 (EDT)

Renamed as "Field Norm of Complex Number", and will add some cautionary words in the "also known as" section. --prime mover (talk) 15:53, 4 April 2019 (EDT)