Determining which of the following Quotients is a Field

Asked Apr 09 '17 at 02:23

Active Apr 09 '17 at 02:23

Viewed 72 times

I took the GRE Math Subject today and I had a question regarding quotients. Let $I = (X^2 + 2)$. For $F = \mathbb{Z}_3, \mathbb{Z}_5, \mathbb{Z}_7, \mathbb{Q}, \mathbb{R}$, determine which fields satisfy the condition that $F[X]/I$ is a field.

Since each answer choice only listed one field, I figured that the first four were in their own category while $\mathbb{R}$ was different so that $\mathbb{R}$ was the correct choice. I am thinking that my answer was correct because $X^2 + 2$ is separable over $\mathbb{R}$.

asked Apr 09 '17 at 02:23

Yunus Syed

1,767

1

$F[X]/I$ is a field $\iff x^2+2$ is irreducible in $F$. Also $x^2+2$ is irreducible in $F\iff $ it has no roots in $F$ – Learnmore Apr 09 '17 at 02:37
Now just check which $F$ satisfies the above and you are done! – Learnmore Apr 09 '17 at 02:38
@RishabhPant isn't $\mathbb{R}[X]/I = \mathbb{R}$? – Mr. Cooperman Apr 09 '17 at 02:41
No ,@Timkinsella; $\Bbb R[X]/I\cong \Bbb R(\alpha)$ where $\alpha^2+2=0$ – Learnmore Apr 09 '17 at 02:46
oh right. ma bad. – Mr. Cooperman Apr 09 '17 at 02:50
So it has no roots in $Z_5, Z_7, Q$ Edit: ....stupid me thought $\sqrt{2}$ was a solution – Yunus Syed Apr 09 '17 at 04:46
GRE Subject Test - Past Papers, Books, Advice – BCLC Apr 16 '17 at 19:34

Determining which of the following Quotients is a Field

0 Answers0