Quotient Ring

**vincisonfire** · November 7th 2008, 03:31 PM

Just to make sure I get it right.
$\mathbb F_5[x]/(x^2+2)$ is a field.
We want to express $(x^2 + x) \cdot (x^3 + 1) - (x + 1)$ as a polynomial of degree less than 2.
$(x^2+2-2+x)\cdot(x^3+2 -2x+1)-(x + 1)$
$(x-2) \cdot (1-2x) - (x + 1)$
$4x - 2x^2-3$
$4x - 2x^2-4+1$
$4x +1$
Thank you!
Also, I have to find the roots of $t^2+3$
$t=\pm\sqrt{-3}=\pm\sqrt{2}$ but 2 is not a perfect square in $\mathbb Z/5\mathbb Z$
$t^2+3$ has no root.
Is that right????

**whipflip15** · November 7th 2008, 07:40 PM

Correct.

Also note that in your field $t^2+3$ reduces to just $1$ .

**vincisonfire** · November 7th 2008, 08:42 PM

But $t^2 +3$ reduce to 1 only if t=x. Therefore there might be roots even if x is not a root. For example, in this field, $t^2 +3$ has two roots that are x and -x.
Yes it's a cheap example but it's late and it still gives the idea. x is not a variable anymore in this field. 'Polynomials' becomes roots.
Anyway, good night.

**ThePerfectHacker** · November 8th 2008, 03:30 PM

Originally Posted by vincisonfire

Thank you!
Also, I have to find the roots of $t^2+3$
$t=\pm\sqrt{-3}=\pm\sqrt{2}$ but 2 is not a perfect square in $\mathbb Z/5\mathbb Z$
$t^2+3$ has no root.
Is that right????

The field $\mathbb{F}_5$ can be identified as a subfield of $\mathbb{F}_5/(x^2+2)$ .
If $\alpha = x + (x^2+2)$ then $\alpha^2 + 2 = 0 \implies \alpha^2 = 3$

The polynomial $t^2 + 3 \in \mathbb{F}_5[t]$ has a root in $\mathbb{F}_5/(x^2+2)$ . Consider $\pm 2\alpha$ . Then $( \pm 2 \alpha)^2 = 4 \alpha^2 = 4\cdot 3 = 2$ . Therefore, $(\pm 2 \alpha)^2 + 3 = 0$ . Which means $2\alpha,-2\alpha$ are its roots.

**luzerne** · November 9th 2008, 11:22 AM

Another (similar) way is to see that the discriminant $b^2-4ac=3$ is a square in the field since $x^2+2=0$ --> $x^2=3$

Then I used the usual formula for quadratic functions and got $t = (\pm\sqrt{3})/2 = \pm x * 3$ (since 1/2 = 3 in Z/5Z), which is the same answer as TPH since $\pm 3 = \pm 2$ in this field

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