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 Gelfond–Schneider theorem, DYK Fact #583
JaneFairfax
Posted: 15:40 Wednesday 05 August 2009


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Did you know? The Gelfond–Schneider theorem states if α is an algebraic number not equal to 0 or 1 and β is an irrational algebraic number, then αβ is transcendental. smile.gif

A proof of this theorem is included in the chapter on transcendental numbers in the book A Course in Number Theory by H.E. Rose. ThumbsUp.gif

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algebraic topology
Posted: 20:28 Wednesday 05 August 2009


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Did you know?

This theorem is a resolution of part of the seventh of the mathematical problems famously proposed by David Hilbert in 1900.

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Ebudae
Posted: 20:44 Wednesday 05 August 2009


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Did you know? Schneider is the German word for a tailor. Waggish.gif



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Ebudæ
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algebraic topology
Posted: 21:48 Wednesday 05 August 2009


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Did you know?

The theorem was first proved by Aleksandr Gelfond in 1934; Theodor Schneider refined the proof in 1935.

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JaneFairfax
Posted: 12:20 Thursday 06 August 2009


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Did you know? The proof of the Gelfond–Schneider theorem given in the book by Rose goes by assuming to the contraray that α, β and αβ all belong to some algebraic-number field K and then constructing a member of K with an inconsistent norm. cool.gif

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algebraic topology
Posted: 11:49 Thursday 27 August 2009


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If F is a subfield of E, a number α in E is said to be algebraic over F iff α is a root of a nonzero polynomial with coefficients in F; α is said to be transcendental over F iff it is not algebraic over F. The Gelfond–Schneider theorem is a result concerning complex transcendental numbers over the rationals.

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JaneFairfax
Posted: 17:00 Thursday 27 August 2009


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Did you know? A complex number is a root of a polynomial with rational coefficients if and only if it is a root of a polynomial with integer coefficients. This is Gauß’s lemma. Clapping.gif

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algebraic topology
Posted: 18:19 Thursday 27 August 2009


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If R is an integral domain with field of fractions F, an element in some field containing F is a root of a polynomial with coefficients in F if and only if it is a root of a polynomial with coefficients in R. For if α is a root of where , and , for , then, if , we have that α is a root of the polynomial where for . The other implication is trivial.

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JaneFairfax
Posted: 09:58 Friday 28 August 2009


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Did you know? If F is a subfield of E and is algebraic over F, the (unique) monic polynomial in F[x] of least degree for which α is a root is called the minimal polynomial of α over F. It is a principal-ideal generator of the kernel of the ring homomorphism , .

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algebraic topology
Posted: 16:29 Friday 28 August 2009


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Did you know?

If F is a subfield of E, the elements of E which are algebraic over F form a subfield of E containing F.

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