Exercise 5.1.13 (Degree Inequality)

Question

Give an example of to show that the in equalityin Corollary 5.1.12 can be strict. Your example can be as trivial as you like.

Hassaan Naeem · Accepted Answer

extbf{Solution:}
We choose our fields and hence extensions to be $\mathbb{C}:\mathbb{R}:\mathbb{Q}$. We also choose $\beta = \sqrt2 \in \mathbb{C}$.
The minimal polynomial of $\sqrt2$ over $\mathbb{Q}$ is $m = t^2 - 2$, 
then deg$_\mathbb{Q}(\beta) = [\mathbb{Q}(\beta):\mathbb{Q}] = 2$.\\
Similarly, the minimal polynomial of $\sqrt2$ over $\mathbb{R}$ is $m = t - \sqrt2$, 
then deg$_{\mathbb{R}}(\beta) = [\mathbb{R}(\beta):\mathbb{R}] = 1$.\\
Hence we have that $[\mathbb{R}(\beta):\mathbb{R}] < [\mathbb{Q}(\beta):\mathbb{Q}] \quad \square$

Exercise 5.1.13 (Degree Inequality)

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Exercise 5.1.13 (Degree Inequality)

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