Trace of an algebraic integer

Definition

The trace of an algebraic integer is the sum of its conjugates, which are the roots of its minimal polynomial over the integers. This concept is crucial in understanding the structure of number fields, as it relates to the behavior and properties of algebraic integers. The trace can provide insights into how elements within a number field interact with one another, especially in terms of their relationships to other algebraic integers and the overall field structure.

5 Must Know Facts For Your Next Test

1. The trace is denoted as Tr(α) for an algebraic integer α, and it is equal to the sum of all conjugates of α.
2. In a number field, the trace can be computed from the coefficients of the minimal polynomial by Vieta's formulas.
3. The trace has important implications for determining whether certain algebraic integers are integral or not within a number field.
4. The trace function is linear, meaning that for any two algebraic integers α and β, Tr(α + β) = Tr(α) + Tr(β).
5. For an algebraic integer α, if its trace is zero, this can suggest that α is related to certain symmetry in its conjugates.

Review Questions

• How does the trace relate to the properties of algebraic integers in a number field?
  • The trace provides valuable information about the relationships among algebraic integers within a number field. Since it sums the conjugates of an algebraic integer, it can reveal symmetries and structures among those elements. Additionally, understanding the trace helps in identifying integral elements and assessing their roles in various arithmetic properties, such as determining bases for rings of integers in number fields.
• Discuss how you would compute the trace of an algebraic integer given its minimal polynomial.
  • To compute the trace of an algebraic integer using its minimal polynomial, you first identify the polynomial's roots, which are the conjugates. By Vieta's formulas, the trace is then calculated as the negative coefficient of the term with degree one less than the leading term (monic), divided by the leading coefficient. This effectively gives you the sum of all roots, allowing you to determine the trace directly from the polynomial coefficients.
• Evaluate the implications of having a zero trace for an algebraic integer and how this affects its conjugates.
  • A zero trace for an algebraic integer implies that the sum of its conjugates cancels out to zero. This condition often indicates specific symmetry among the conjugates, which can lead to interesting structural properties within the number field. For instance, if an algebraic integer has a zero trace, this might suggest that it lies at a special location in relation to other integers in the field or that it could play a role in certain factorization patterns or units within that field.

Related terms

Algebraic Integer:

An algebraic integer is a complex number that is a root of a monic polynomial with integer coefficients.

Conjugate: A conjugate of an algebraic integer is another root of its minimal polynomial, revealing symmetric properties of the roots.

Minimal Polynomial:

The minimal polynomial of an algebraic integer is the monic polynomial of lowest degree that has the integer as a root, and it uniquely determines the algebraic integer's properties.