Field Product with Zero

Theorem

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

Let $a \in F$.

Then:

$a \times 0_F = 0_F$

That is, $0_F$ acts as a zero element of $F$, which justifies its name.

$\ds a$

$=$

$\ds a \times 1_F$

Field Axiom $\text M3$: Identity for Product: $1_F$ is the unity of $F$

$\ds $

$=$

$\ds a \times \paren {0_F + 1_F}$

Field Axiom $\text A3$: Identity for Addition

$\ds $

$=$

$\ds a \times 0_F + a \times 1_F$

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

$\ds $

$=$

$\ds a \times 0_F + a$

Field Axiom $\text M3$: Identity for Product

$\ds \leadsto \ \ $

$\ds \paren {a \times 0_F + a} + \paren {-a}$

$=$

$\ds a + \paren {-a}$

adding $-a$ to both sides

$\ds \leadsto \ \ $

$\ds a \times 0_F + \paren {a + \paren {-a} }$

$=$

$\ds a + \paren {-a}$

Field Axiom $\text A1$: Associativity of Addition

$\ds \leadsto \ \ $

$\ds a \times 0_F + 0_F$

$=$

$\ds 0_F$

Field Axiom $\text A4$: Inverses for Addition

$\ds \leadsto \ \ $

$\ds a \times 0_F$

$=$

$\ds 0_F$

Field Axiom $\text A3$: Identity for Addition

$\blacksquare$

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