[Note:I have yet not studied field theory,I am just using it in vector spaces.So my question may look silly.] Suppose $F$ is a field and $F \subset \mathbb C$.Then does $F^3$ contain $(1,0,0),(0,1,0), (0,0,1)$. Can anyone show me if they always exist in $F$ and if yes then how?
2026-04-08 13:12:16.1775653936
Suppose $F$ is a field and $F \subset \mathbb C$.Then does $F^3$ contain $(1,0,0),(0,1,0), (0,0,1)$.
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A field $F$ contains a neutral element for multiplication and one for addition, say $0$ is the neutral element of addition and $1$ the neutral element of multiplication. So $(1,0,0)$ is in $F^3$ since $1 \in F$ and $0 \in F$. Same argument for the other two vectors.