Field has 2 Ideals
Jump to navigation
Jump to search
Theorem
Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$.
Then the only ideals of $\struct {F, +, \circ}$ are $\struct {F, +}$ and $\set {0_F}$.
That is, $\struct {F, +, \circ}$ has no non-null proper ideals.
Proof
By definition, a field is a division ring.
From Null Ring is Ideal and Ring is Ideal of Itself, it is always the case that $\set {0_F}$ and $\struct {F, +}$ are ideals of $\struct {F, +, \circ}$.
From Ideals of Division Ring, it follows that the only ideals of $\struct {F, +, \circ}$ are $\struct {F, +}$ and $\set {0_F}$.
$\blacksquare$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 21$. Ideals: Theorem $36$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 58.3$ Ideals