Problem 10 - A Course in Galois Theory

Question

What are the units in \(R[x]\), where \(R\) is an integral domain? What are the units in \(\mathbb{Z}_{4}[x]\) ?

Step-by-Step Solution

Answer

Units in \(R[x]\) are constant polynomials with non-zero units in \(R\). Units in \(\mathbb{Z}_{4}[x]\) are polynomials with coefficients 1 or 3.

Step 1: Understanding Units in Polynomial Rings

Units in a ring are elements that have a multiplicative inverse within the same ring. In the polynomial ring \( R[x] \), a polynomial is a unit if there exists another polynomial such that their product is equal to 1, the multiplicative identity in the ring.

Step 2: Identifying Units in \( R[x] \)

In \( R[x] \), where \( R \) is an integral domain, a polynomial \( f(x) \) is a unit if and only if it has an inverse in \( R[x] \). This happens when \( f(x) = a \), where \( a \) is a non-zero unit in \( R \), and in this case, only constant polynomials consisting of units of \( R \) are units in \( R[x] \).

Step 3: Understanding Units in \( \mathbb{Z}_{4} \)

\( \mathbb{Z}_{4} \) is not an integral domain, but we still apply similar logic for discovering units. An element in \( \mathbb{Z}_{4} \) is a unit if it has a multiplicative inverse. Review the elements: 0 has no inverse, 1 and 3 are units because \(1 \times 1 = 1 \mod 4\) and \(3 \times 3 = 9 \equiv 1 \mod 4\). Therefore, 1 and 3 have inverses in \( \mathbb{Z}_{4}.\)

Step 4: Determining Units in \( \mathbb{Z}_{4}[x] \)

In \( \mathbb{Z}_{4}[x] \), a polynomial is a unit if its leading coefficient is a unit in \( \mathbb{Z}_{4}\). The possible units are constant polynomials with coefficients 1 or 3. Therefore, the units in \( \mathbb{Z}_{4}[x] \) are any polynomials whose leading and all other coefficients are either 1 or 3.

Key Concepts

Integral Domain

An integral domain is a special type of ring that is crucial in understanding polynomial rings and their units. It is a commutative ring with no zero divisors, meaning the multiplication of any two non-zero elements does not yield zero. This property ensures a form of 'cancellation law' in the ring.

• For example, the ring of integers, \(\mathbb{Z}\), is an integral domain, because if \((a \times b = 0)\), then either \(a = 0\) or \(b = 0\).
• This property is essential for defining concepts like multiplicative inverses and units, because it guarantees that the cancellation laws hold.

Understanding integral domains is key when determining the units in polynomial rings, as it lays the foundational properties that these elements must obey.

Polynomial Rings

Polynomial rings, denoted as \(R[x]\), are extensions of rings where the elements are polynomials with coefficients from the ring \(R\). In a polynomial ring, the operations of addition and multiplication are defined just like in numerical polynomials, using the coefficients and the power of each term.

• Each element in a polynomial ring has a degree, which is the highest power of \(x\) with a non-zero coefficient.
• The leading coefficient is the coefficient of the highest degree term and plays a crucial role in determining properties such as invertibility.

In the context of polynomial rings, a unit is a polynomial that has an inverse, typically requiring that its leading coefficients are units of the original ring \(R\). This becomes a fundamental concept when discussing the units in polynomial rings like \(R[x]\).

Multiplicative Inverse

The multiplicative inverse in the context of polynomial rings is an important concept, especially when considering units. The multiplicative inverse of an element \(a\) is an element \(b\) such that their product equals the multiplicative identity, which is usually 1 in rings.

• For an element to have a multiplicative inverse, it means there must be another element in the same ring that can multiply it to yield 1.
• In polynomial rings \(R[x]\), only constant polynomials whose coefficients are units in \(R\) have inverses.

This concept extends to determining which polynomials in a ring are units, as it depends on whether the polynomial can be reduced to a form whose inverse can be easily identified.

Constant Polynomials

Constant polynomials are polynomials consisting entirely of a constant term, i.e., the degree is zero. When we talk about units in polynomial rings, constant polynomials take a prominent role because of their structure.

• These are particularly interesting because if the constant is a unit in the ring \(R\), the entire polynomial is a unit in \(R[x]\).
• If the leading coefficient is a unit in \(R\), the constant polynomial itself is invertible, making it a key player in the definition of units in polynomial rings.

The attributes of constant polynomials simplify determining which polynomials in a ring can have a multiplicative inverse, especially in integral domains or other ring structures like \(\mathbb{Z}_4\).