2025-01-19

967: For Field, Positive-Natural-Number-th Root of 0 Is 0

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description/proof of that for field, positive-natural-number-th root of 0 is 0

Topics


About: field

The table of contents of this article


Starting Context



Target Context


  • The reader will have a description and a proof of the proposition that for any field, any positive-natural-number-th root of 0 is 0.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Structured Description


Here is the rules of Structured Description.

Entities:
F: { the fields }
n: N{0}
R0,n: ={αF|αn=0}
//

Statements:
R0,n={0}
//


2: Proof


Whole Strategy: Step 1: see that 0R0,n; Step 2: see that for each αR0,n, α=0.

Step 1:

Let us see that 0n=0.

For each rF, r0=0: r(1+0)=r1=r, because 0 is the additive identity and 1 is the multiplicative identity, but r(1+0)=r1+r0, by the distributability, =r+r0, because 1 is the multiplicative identity; so, r+r0=r, and r0=r+r+r0=r+r=0.

Especially, 02=00=0.

Supposing 0m=0, 0m+1=0m0=02=0.

So, by the induction principle, 0n.

Step 2:

Let αR0,n be any.

αn=0.

When n=1, α1=α=0.

Let us suppose that 1<n hereafter.

Let us suppose that α0. Then, there would be a multiplicative inverse, α1F. αn1=αnα1=0α1=0. When n1=1, αn1=α=0, a contradiction. Otherwise, αn2=0, likewise. When n2=1, α=0, a contradiction. Otherwise, ..., etc. Anyway, eventually, α=0, a contradiction.

So, α=0.


References


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