Field Extension/Examples/Complex Numbers over Reals
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Examples of Field Extensions
The complex numbers $\C$ forms a finite field extension over the real numbers $\R$ of degree $2$.
Proof
It follows from Real Numbers form Subfield of Complex Numbers that $\C$ is an extension of $\R$.
From Vector Space over Division Subring: Real Numbers in Complex Numbers we have that the dimension of the vector space on $\C$ over $\R$ is $2$.
$\blacksquare$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Field Extensions: $\S 36$. The Degree of a Field Extension: Example $69$