Elements of Abstract Algebra by Allan Clark

Author:Allan Clark
Language: eng
Format: epub, pdf
ISBN: 9780486140353
Publisher: Dover Publications
Published: 2012-07-24T16:00:00+00:00

112β. Prove that if α and β are transcendental numbers, then either α + β or αβ is transcendental.

112γ. Let φ be a polynomial over Q. Show that φα = n implies that α is transcendental.

113. Proposition. Every irreducible polynomial over a number field has distinct roots.

Proof. Let ƒ be an irreducible polynomial over a number field F given by

ƒx = c0 + c1x + ··· + cnxn.

Suppose ƒ has a multiple root α. Then over C we can write

ƒx = cn(x − α)m(x − α1) ··· (x − αn −m),

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