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Mathematics
Linear and Abstract Algebra
Splitting Fields - Example 3 - D&F Section 13.4, pages 537 -
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[QUOTE="Math Amateur, post: 5762464, member: 203675"] I am reading Dummit and Foote, Chapter 13 - Field Theory. I am currently studying Section 13.4 : Splitting Fields and Algebraic Closures ... ... I need some help with an aspect of Example 3 of Section 13.4 ... ... Example 3 reads as follows: [ATTACH=full]203548[/ATTACH] [ATTACH=full]203549[/ATTACH] In the above text by Dummit and Foote, we read the following: " ... ... Since ##\sqrt{ -3 }## satisfies the equation ##x^2 + 3 = 0## the degree of this extension over ##\mathbb{Q} ( \sqrt [3] {2} )## is at most ##2##, hence must be ##2## since we observed above that ##\mathbb{Q} ( \sqrt [3] {2} )## is not the splitting field ... ... "I do not understand why the degree of the extension ##K## over ##\mathbb{Q} ( \sqrt [3] {2} )## must be exactly ##2## ... ... why does ##\mathbb{Q} ( \sqrt [3] {2} )## not being the splitting field ensure this ... ... Can someone please give a simple and complete explanation ... Hope someone can help ... Peter [/QUOTE]
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Mathematics
Linear and Abstract Algebra
Splitting Fields - Example 3 - D&F Section 13.4, pages 537 -
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