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- #1

- Jun 22, 2012

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Example 15 on page 282 (see attachment) reads as follows:

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**Example 15.**Let [TEX] E = \mathbb{Q} ( \sqrt{2} , \sqrt{5} ) [/TEX].

Find [TEX] [E \ : \ \mathbb{Q} ] [/TEX] , exhibit a [TEX] \mathbb{Q} [/TEX]-basis of E, and show that [TEX] E = \mathbb{Q} ( \sqrt{2} + \sqrt{5} ) [/TEX]. Then find the minimum polynomial of [TEX] \sqrt{2} + \sqrt{5} [/TEX] over [TEX] \mathbb{Q} [/TEX].

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**In the solution we read:**

*We write [TEX] L = \mathbb{Q} ( \sqrt{2} ) [/TEX] for convenience so that [TEX] E = L(\sqrt{5}) [/TEX] ... ... etc*

**Solution:**... ... ... We claim that [TEX] X^2 - 5 [/TEX] is the minimal polynomial of [TEX] \sqrt{5} [/TEX] over L. Because [TEX] \sqrt{5} [/TEX] and [TEX] - \sqrt{5} [/TEX] are the only roots of [TEX] X^2 - 5 [/TEX] in [TEX] \mathbb{R} [/TEX], we merely need to show that [TEX] \sqrt{5} \notin L [/TEX]. ... ... etc

**My problem is the following:**

How does showing [TEX] \sqrt{5} \notin L [/TEX] imply that [TEX] X^2 - 5 [/TEX] is the minimal polynomial of [TEX] \sqrt{5} [/TEX] over L?

Can someone please help with this issue?

Peter