Question

Let \(F\) be a field of characteristic other than \(2 .\) Show that a non-zero \(g \in F\left(\left(X^{-1}\right)\right)\) has a square-root in \(F\left(\left(X^{-1}\right)\right)\) if and only if \(\operatorname{deg}(g)\) is even and \(\operatorname{lc}(g)\) has a square-root in \(F\).

Short answer

Given a field \(F\) with characteristic not equal to \(2\) and a nonzero element \(g\) in \(F\left(\left(X^{-1}\right)\right)\), provide the conditions for \(g\) to have a square root in \(F\left(\left(X^{-1}\right)\right)\). Answer: \(g\) has a square root in \(F\left(\left(X^{-1}\right)\right)\) if and only if the degree of \(g\) is even, and the leading coefficient of \(g\) has a square root in \(F\).

Step-by-step solution

Necessity: If \(g\) has a square root, then the degree of \(g\) is even, and the leading coefficient of \(g\) has a square root in \(F\).

Let \(g\) have a square root in \(F\left(\left(X^{-1}\right)\right)\). Let \(h\) be the square root of \(g\), so \(h^2 = g\). Since \(g\) and \(h\) are both elements of \(F\left(\left(X^{-1}\right)\right)\), we can write them as rational functions of the form: $$g=\frac{a(X)}{b(X)} \quad \text{and} \quad h=\frac{c(X)}{d(X)}$$ where \(a(X), b(X), c(X), d(X) \in F[X]\). Then, we have that $$\frac{a(X)}{b(X)} = h^2 = \frac{c^2(X)}{d^2(X)}.$$ This means that \(a(X)d^2(X) = b(X)c^2(X)\). Now consider the degree of both sides: $$\operatorname{deg}(a(X)) + 2\operatorname{deg}(d(X)) = \operatorname{deg}(b(X)) + 2\operatorname{deg}(c(X))$$ Both instances of 2 times a degree must be even integers, since they are doubling the degree. Therefore, \(\operatorname{deg}(a(X))-\operatorname{deg}(b(X))\) is also even, which is the degree of \(g\). So, the degree of \(g\) is even. Additionally, squaring both sides of \(h^2 = g\) results in their leading coefficients also being squared. Let \(l\) and \(m\) be the leading coefficients of \(g\) and \(h\), respectively. Then, \(l = m^2\) and it follows that \(l\) has a square root in \(F\).

Sufficiency: If the degree of \(g\) is even and the leading coefficient of \(g\) has a square root in \(F\), there exists a square root of \(g\) in \(F\left(\left(X^{-1}\right)\right)\).

Suppose \(\operatorname{deg}(g)\) is even, and \(\operatorname{lc}(g)\) has a square root in \(F\). Let \(g = \frac{a(X)}{b(X)}\) with \(a(X), b(X) \in F[X]\). For \(h\) to be a square root, we must find \(c(X), d(X) \in F[X]\) such that $$g=\frac{a(X)}{b(X)} = \left(\frac{c(X)}{d(X)}\right)^2 = \frac{c^2(X)}{d^2(X)}.$$ Since the degree of the numerator and denominator must balance, this implies $$\operatorname{deg}(a(X)) = 2\operatorname{deg}(c(X)) \quad \text{and} \quad \operatorname{deg}(b(X)) = 2\operatorname{deg}(d(X)).$$ Choose \(c(X)\) and \(d(X)\) such that these conditions are satisfied. Also, we must find \(c(X)\) and \(d(X)\) such that their leading coefficients are square roots of their respective leading coefficients of \(a(X)\) and \(b(X)\). This is possible because \(\operatorname{lc}(g)\) has a square root in \(F\) and the leading coefficients of \(a(X)\) and \(b(X)\) have the same square root. Now, we have an \(h\) of the form $$h = \frac{c(X)}{d(X)}$$ that satisfies \(h^2 = g\). Thus, there exists a square root of \(g\) in \(F\left(\left(X^{-1}\right)\right)\).

Key concepts

Characteristic of a Field

The characteristic of a field is a fundamental concept in field theory. It is the smallest positive integer 'n' such that the sum of 'n' copies of the field's multiplicative identity (1) is equal to the additive identity (0). In formula terms, it is the smallest positive integer 'n' for which \( n \times 1 = 0 \). If no such positive integer exists, the field is said to have characteristic zero, which implies that it contains a copy of the infinite field of rational numbers, \( \textbf{Q} \).

Understanding the characteristic is crucial when studying field extensions and considering algebraic properties like solving polynomial equations, since different characteristics lead to notably different behavior of algebraic structures. For instance, a field with characteristic other than 2 ensures that the equation \( x^2 = a \) has at most two solutions for any 'a' in the field, providing a clearer path to finding square roots within the field.

Square-Root in Fields

When we talk about the square root of an element in a field, we're looking for another element within the same field that, when squared, gives us the original element. The operation of taking a square root, denoted \( \text{sqrt}(x) \) or \( x^{1/2} \), is more complex in fields than in real numbers because not every element may have a square root within the field.

In fields of characteristic other than 2, like the one from our exercise, finding square roots can be less complicated because the polynomial \( x^2 - a \) will not suffer from the issues encountered in fields of characteristic 2. However, if a field's characteristic is 2, every element is its own square root, as \( x^2 = x \) for all 'x' in the field. This uniqueness makes working with square roots in such fields quite different and requires careful attention.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial that has a non-zero coefficient. It gives us vital information about the polynomial's behavior, particularly in the context of root-finding and graphing.

For the polynomial \( f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \), where \( a_n \) is non-zero, the degree is 'n'. For example, in \( x^3 - 2x + 5 \), the degree is 3. This concept also tells us about the maximum number of roots a polynomial can have, which is equal to its degree, an essential fact used in analyzing solutions to polynomial equations. In the context of fields, the degree of a polynomial plays a key role in determining properties like the existence of square roots, as seen in the exercise.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. In the polynomial \( f(x) = a_n x^n + ... + a_1 x + a_0 \), where \( a_n \) is non-zero, \( a_n \) is the leading coefficient. It's an essential ingredient in determining the end behavior of a polynomial's graph.

For instance, if we have the polynomial \( 3x^2 + 2x - 5 \), the leading coefficient is 3. This coefficient dictates that as 'x' goes to positive or negative infinity, the polynomial's behavior will predominantly be influenced by the term \( 3x^2 \). When we square a polynomial or find roots of polynomials as in the exercise, the leading coefficient may determine whether such operations are possible within the given field.