xcubedminus2

# xcubedminus2 - Math 220C Galois theory of x 3 2 We want to...

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Unformatted text preview: Math 220C April 11, 2008 Galois theory of x 3- 2 We want to construct the splitting field for x 3- 2 over Q , and explicitly find its Galois group and the intermediate subfields. Since x 3- 2 is irreducible in Q [ x ], adjoining a root will give a field which is iso- morphic to Q [ x ] / ( x 3- 2). If we let 3 √ 2 denote this root, then the extension field is Q ( 3 √ 2). We can factor x 3- 2 in Q ( 3 √ 2)[ x ] as follows: x 3- 2 = ( x- 3 √ 2)( x 2 + 3 √ 2 x + 3 √ 4) . The polynomial x 2 + 3 √ 2 x + 3 √ 4 ∈ Q ( 3 √ 2)[ x ] is irreducible, because x 3- 2 can only have a single root in Q ( 3 √ 2) (since this field can be embedded in R , in which x 3- 2 has only a single root). Thus, the second step in constructing the splitting field consists of introducing a root a of x 2 + 3 √ 2 x + 3 √ 4 over Q ( 3 √ 2), which satisfies a 2 =- a 3 √ 2- 3 √ 4; the splitting field is then Q ( 3 √ 2)( a ). The complete factorization over Q ( 3 √ 2)( a ) is x 3- 2 = ( x-...
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xcubedminus2 - Math 220C Galois theory of x 3 2 We want to...

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