# Definition talk:Field Norm of Complex Number

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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)