91Ó°ÊÓ

An extension field is a larger field that contains a smaller field as a subfield. In our example, \(\mathbb{Q}(\sqrt{2})\) is an extension field of \(\mathbb{Q}\). What this means is: we start with the field of rational numbers (\(\mathbb{Q}\)), and then include all numbers that can be formed by adding and multiplying \(\sqrt{2}\) with the numbers in \(\mathbb{Q}\). Consequently, any number in the form \(a + b\sqrt{2}\) (with \(a, b\) rational) is included in \(\mathbb{Q}(\sqrt{2})\).

• The origin field here is \(\mathbb{Q}\), or the set of all rational numbers.
• By adjoining \(\sqrt{2}\), we create the extension field \(\mathbb{Q}(\sqrt{2})\).
• This extension allows solutions to polynomial equations, which \(\mathbb{Q}\) alone cannot resolve.

Understanding such extensions helps solve equations that would otherwise not have solutions in the defined origin field. Fields give a structured way of including new elements like roots of polynomials.

A subfield is a field contained within another field, such that all operations (addition, multiplication, etc.) performed in the subfield are valid when considered as operations in the larger field. \(\mathbb{Q}\) is a subfield of \(\mathbb{Q}(\sqrt{2})\) because any rational number expressed as \(q + 0\sqrt{2}\) is still part of \(\mathbb{Q}(\sqrt{2})\).
This subfield relationship means:

• All elements of \(\mathbb{Q}\) can be expressed as elements of \(\mathbb{Q}(\sqrt{2})\).
• The operations defined on \(\mathbb{Q}\) naturally extend to \(\mathbb{Q}(\sqrt{2})\).
• \(\mathbb{Q}\) maintains its field properties even within \(\mathbb{Q}(\sqrt{2})\).

Hence, any properties or solutions that hold in \(\mathbb{Q}\) are automatically carriedover to \(\mathbb{Q}(\sqrt{2})\), maintaining consistency across the fields.

Quadratic equations are polynomials of the form \(ax^2 + bx + c = 0\). Solving these equations often involves finding values of \(x\) that satisfy the equation, and these values are known as roots. The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is commonly used.
In our exercises, examining if the roots of specific quadratic equations lie in \(\mathbb{Q}(\sqrt{2})\) is important.

• For the equation \(x^2 - 4x + \frac{7}{2} = 0\), the roots \(2.5\) and \(1.5\) are clearly rational and thus in \(\mathbb{Q}(\sqrt{2})\).
• The equation \(x^2 - 4x + 3 = 0\) gives roots \(3\) and \(1\), both rational, and thus also in \(\mathbb{Q}(\sqrt{2})\).

These exercises demonstrate how the extension field \(\mathbb{Q}(\sqrt{2})\) contains all solutions to certain quadratic equations, making it useful for solving extensions of number systems.

Fields have specific properties: closure, associativity, commutativity, identity, and inverses under both addition and multiplication. These are necessary for a set to be a field.

• Closure: For any elements \(a+b\sqrt{2}\) and \(c+d\sqrt{2}\) in \(\mathbb{Q}(\sqrt{2})\), both their sum and product remain in \(\mathbb{Q}(\sqrt{2})\).
• Associativity and Commutativity: These hold as expected due to the properties of rational numbers, from which \(\mathbb{Q}(\sqrt{2})\) is derived.
• Identity: The additive identity is \(0 + 0\sqrt{2}\), while the multiplicative is \(1 + 0\sqrt{2}\).
• Inverses: Any non-zero element \(a+b\sqrt{2}\) has an inverse \(\frac{a-b\sqrt{2}}{a^2-2b^2}\), ensuring division is always possible.

Confirming these properties for \(\mathbb{Q}(\sqrt{2})\) defines it as a field, allowing robust mathematical operations within the set.