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Replaced vague "algebraic integers" with what was actually meant

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edited Jan 25, 2013 at 2:50

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Let x be an algebraic number of unspecified degree, expressed using arithmetic, rational powers, and algebraic integers~~algebraic integers~~ (edit: Root[...] constructs). I would like to test conclusively whether it is zero.

I don't know whether any of the following are guaranteed to work:

x==0
Simplify[x]==0
FullSimplify[x]==0
PossibleZeroQ[x,Method->"ExactAlgebraics"]

The following should work, but seems unlikely to be efficient:

MinimalPolynomial[x][y]===y

I don't want to use numerical approximation unless the answer is guaranteed to be correct.

Do any of the first four lines above guarantee a correct answer?

What is the best way to perform this test in Mathematica?

Let x be an algebraic number of unspecified degree, expressed using arithmetic, rational powers, and algebraic integers. I would like to test conclusively whether it is zero.

I don't know whether any of the following are guaranteed to work:

x==0
Simplify[x]==0
FullSimplify[x]==0
PossibleZeroQ[x,Method->"ExactAlgebraics"]

The following should work, but seems unlikely to be efficient:

MinimalPolynomial[x][y]===y

I don't want to use numerical approximation unless the answer is guaranteed to be correct.

Do any of the first four lines above guarantee a correct answer?

What is the best way to perform this test in Mathematica?

Let x be an algebraic number of unspecified degree, expressed using arithmetic, rational powers, and ~~algebraic integers~~ (edit: Root[...] constructs). I would like to test conclusively whether it is zero.

I don't know whether any of the following are guaranteed to work:

x==0
Simplify[x]==0
FullSimplify[x]==0
PossibleZeroQ[x,Method->"ExactAlgebraics"]

The following should work, but seems unlikely to be efficient:

MinimalPolynomial[x][y]===y

I don't want to use numerical approximation unless the answer is guaranteed to be correct.

Do any of the first four lines above guarantee a correct answer?

What is the best way to perform this test in Mathematica?

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asked Jan 24, 2013 at 20:22

Tobias Hagge

1.4k
9
17

Most efficient way to determine conclusively whether an algebraic number is zero

Let x be an algebraic number of unspecified degree, expressed using arithmetic, rational powers, and algebraic integers. I would like to test conclusively whether it is zero.

I don't know whether any of the following are guaranteed to work:

x==0
Simplify[x]==0
FullSimplify[x]==0
PossibleZeroQ[x,Method->"ExactAlgebraics"]

The following should work, but seems unlikely to be efficient:

MinimalPolynomial[x][y]===y

I don't want to use numerical approximation unless the answer is guaranteed to be correct.

Do any of the first four lines above guarantee a correct answer?

What is the best way to perform this test in Mathematica?

algebraic-manipulation