Prove that an algebraic integer is a unit in Z(a)

Thread starter lahuxixi
Start date Mar 7, 2013
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In summary, an algebraic integer is a complex number that satisfies a monic polynomial equation with integer coefficients. It can be a unit in Z(a) if it has a multiplicative inverse in the ring of integers of the algebraic number field a. To prove this, one must show that it has a multiplicative inverse in the ring of integers using methods such as the Extended Euclidean Algorithm. However, not all algebraic integers are units in Z(a) for all algebraic number fields a, as the concept of a unit in Z(a) depends on the specific algebraic number field and its ring of integers. Additionally, not all units in Z(a) are algebraic integers, as some may be complex numbers that do not satisfy

Mar 7, 2013

lahuxixi

I'm completely stuck here, can anyone help me?

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Mar 7, 2013

Dick

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lahuxixi said:

I'm completely stuck here, can anyone help me?

You shouldn't be completely stuck. One direction at least is easy. ##\alpha^n+a_{n-1}\alpha^{n-1}+...+a_1 \alpha=-a_0##. Can you show that if ##a_0=\pm 1## then ##\alpha## is a unit?

Related to Prove that an algebraic integer is a unit in Z(a)

1. What is an algebraic integer?

An algebraic integer is a complex number that satisfies a monic polynomial equation with integer coefficients. In other words, it is a root of a polynomial with integer coefficients.

2. What does it mean for an algebraic integer to be a unit in Z(a)?

An algebraic integer is a unit in Z(a) if it has a multiplicative inverse in the ring of integers of the algebraic number field a. In simpler terms, it means that the algebraic integer can be multiplied by another integer to equal 1.

3. How do you prove that an algebraic integer is a unit in Z(a)?

To prove that an algebraic integer is a unit in Z(a), you must show that it has a multiplicative inverse in the ring of integers of the algebraic number field a. This can be done by finding the inverse using the Extended Euclidean Algorithm or other methods.

4. Can an algebraic integer be a unit in Z(a) for all algebraic number fields a?

No, not all algebraic integers are units in Z(a) for all algebraic number fields a. The concept of a unit in Z(a) depends on the specific algebraic number field and the properties of its ring of integers.

5. Are all units in Z(a) algebraic integers?

No, not all units in Z(a) are algebraic integers. Some units may be complex numbers that are not roots of a polynomial with integer coefficients, and therefore do not satisfy the definition of an algebraic integer.