# Field if I define The plane

1. Jan 30, 2013

### Bachelier

if I define The plane: $F = ℝ$ x $ℝ = \{ (a, b) | a, b ∈ ℝ \}$

and define addition and multiplication as:
(a, b) + (c, d) := (a + c, b + d)
(a, b) · (c, d) := (ac, bd)

Then $F$ is a field. right?

would the multiplication as described here make ℂ a field?

2. Jan 30, 2013

### pwsnafu

Re: Field

No, you have zero divisors.

How would you do i2 = -1?

3. Jan 31, 2013

### Bachelier

Re: Field

You mean like (0,7).(8,0)

4. Jan 31, 2013

### HallsofIvy

Staff Emeritus
Yes, neither (7, 0) nor (0, 8) is the additive identity but neither has a mulitplicative inverse.

5. Jan 31, 2013

### Bachelier

thanks. Basically ℂ will fail to be an integral domain in the first place under this operation.

Last edited: Jan 31, 2013