Question

Show that if the factorization of \(f\) over \(F[X]\) into monic irreducibles is \(f=f_{1}^{e_{1}} \cdots f_{r}^{e_{r}},\) and if \(\alpha=[h]_{f} \in F[X] /(f),\) then the minimal polynomial \(\phi\) of \(\alpha\) over \(F\) is \(\operatorname{lcm}\left(\phi_{1}, \ldots, \phi_{r}\right),\) where each \(\phi_{i}\) is the minimal polynomial of \([h]_{f_{i}^{e_{i}}} \in F[X] /\left(f_{i}^{e_{i}}\right)\) over \(F\)

Short answer

Answer: The minimal polynomial of \(\alpha\) is the least common multiple of the set of minimal polynomials \(\{\phi_1, ..., \phi_r\}\) of \(\alpha\) under each irreducible factor \(f_i^{e_i}\).

Step-by-step solution

Minimal Polynomials of \(\alpha\) under each \(f_i^{e_i}\)

First, we need to find the minimal polynomials of \(\alpha\) under each irreducible factor \(f_i^{e_i}\). To do this, let's represent the element \(\alpha\) in each of the factor rings \(F[X]/(f_i^{e_i})\) as \([\alpha]_{f_i^{e_i}}\). Then the minimal polynomial of \(\alpha\) in each factor ring is the unique monic polynomial \(\phi_i\) such that \(\phi_i([\alpha]_{f_i^{e_i}}) = 0\).

LCM of the minimal polynomials

Now that we have the minimal polynomials \(\phi_i\) in each factor ring, we want to find their least common multiple (LCM). Recall that the LCM of a set of polynomials is the polynomial with the smallest degree that each of the given polynomials divides. Let's denote the LCM as \(\operatorname{lcm}\left(\phi_{1}, \ldots, \phi_{r}\right)\).

Proving the LCM is the minimal polynomial of \(\alpha\)

Now we must show that the least common multiple of \(\phi_1, ..., \phi_r\) is the minimal polynomial of \(\alpha\). First, we observe that \(\operatorname{lcm}\left(\phi_{1}, \ldots, \phi_{r}\right)(\alpha) = 0\), since \(\alpha\) is a root of each \(\phi_i\). Now consider any other polynomial \(g \in F[X]\) that has \(\alpha\) as a root (i.e., \(g(\alpha) = 0\)). We must show that the LCM of the minimal polynomials divides \(g\). Since each \(\phi_i\) divides \(g\) (due to \(\alpha\) being a root of every \(\phi_i\)), it follows that their least common multiple also divides \(g\). Thus, we have shown that the least common multiple of the set of minimal polynomials \(\{\phi_1, ..., \phi_r\}\) is the minimal polynomial of \(\alpha\) in \(F[X]/(f)\). Therefore, for \(\alpha = [h]_f \in F[X]/(f)\), the minimal polynomial \(\phi\) of \(\alpha\) over \(F\) is \(\operatorname{lcm}\left(\phi_{1}, \ldots, \phi_{r}\right)\).

Key concepts

Monic Irreducibles

In the world of polynomials, a monic polynomial is one where the leading coefficient is 1. These kinds of polynomials are especially crucial when discussing minimal polynomials, as they provide a standardized form for ease of analysis and computation.

Irreducible polynomials, on the other hand, cannot be factored into simpler polynomials with coefficients in the same field. They are akin to prime numbers within the set of integers.

When we talk about factorizing a polynomial like \(f\) over \(F[X]\) into monic irreducibles, we're looking for a way to break \(f\) down into a product of these minimal factor components. Each factor, \(f_i\), is monic and irreducible, and it is raised to a power, \(e_i\), often representing the polynomial's multiplicity in this factorization.

• A monic polynomial has a leading coefficient of 1.
• An irreducible polynomial has no factors other than itself and 1 over a given field.
• Factoring into monic irreducibles helps simplify problems in polynomial arithmetic and algebraic structures like factor rings.

Factor Rings

Factor rings are captivating structures that arise when we divide a polynomial ring by an ideal defined by a polynomial.

For instance, consider a polynomial \(f\) in \(F[X]\). The factor ring, denoted as \(F[X]/(f)\), comprises all equivalence classes of polynomials modulo \(f\). This means we consider two polynomials equivalent if their difference is divisible by \(f\).

In simpler terms, a factor ring restructures the set of polynomials to focus only on those compatible modulo the chosen polynomial:

• Each element of a factor ring is an equivalence class of polynomials.
• Operations in the factor ring are executed analogously to usual polynomial computations but are followed by taking the remainder when divided by the given polynomial.
• Factor rings provide insights into more complex algebraic structures, like fields or vector spaces, depending on the properties of \(f\).

Least Common Multiple

The concept of the least common multiple (LCM) is instrumental when dealing with polynomials, especially in finding minimal polynomials.

The LCM of a set of polynomials is the smallest polynomial that each member of the set will divide without a remainder. It shares a similarity with the LCM of integers but applied within polynomial arithmetic.

In our context, when determining the minimal polynomial \(\phi\) of an element \(\alpha\) in a factor ring \(F[X]/(f)\), the LCM of the polynomials \(\phi_1, \phi_2, \ldots, \phi_r\)—each of which represents the minimal polynomial under a respective \(f_i^{e_i}\)—gives us this minimal polynomial. This highlights that:

• The LCM is a single polynomial representing all common roots in the least degree achievable.
• Finding the LCM ensures that the union of divisibility properties of its constituents is preserved.
• This LCM uniquely defines the minimal polynomial that \(\alpha\) satisfies.