Product with Field Negative/Corollary

Theorem

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$.

Let $a \in F$.

Then:

$\paren {-1_F} \times a = \paren {-a}$

Proof

\(\ds a + \paren {-1_F} \times a\)

\(=\)

\(\ds 1_F \times a + \paren {-1_F} \times a\)

Field Axiom $\text M3$: Identity for Product

\(\ds \)

\(=\)

\(\ds \paren {1_F + \paren {-1_F} } \times a\)

Field Axiom $\text D$: Distributivity of Product over Addition

\(\ds \)

\(=\)

\(\ds 0_F \times a\)

Field Axiom $\text A4$: Inverses for Addition

\(\ds \)

\(=\)

\(\ds 0_F\)

Field Product with Zero

\(\ds \leadsto \ \ \)

\(\ds \paren {-a}\)

\(=\)

\(\ds \paren {-1_F} \times a\)

Definition of Ring Negative

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $3$: Field Theory: Definition and Examples of Field Structure: $\S 87 \beta$
• 1973: C.R.J. Clapham: Introduction to Mathematical Analysis ... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields: Theorem $2 \ \text {(vi)}$