Understanding Field Extensions and Irreducible Quadratics in Q[x]

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In summary, the conversation is about a student struggling with a homework problem involving field extensions and an irreducible quadratic with one root in the extension. They are trying to find a case where there is only one root in the extension, but are confused because of the quadratic formula. They are considering using the fact that the extension is of degree 2 over Q to solve the problem.
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Homework Statement



I am just learning about field extensions, so I am not as comfortable with them yet as I'd like to be. I have a more general homework problem, but let me post a special case of the part I am focusing on which contains the part that is confusing me.

Suppose I have an extension K/Q of degree 2, where Q is the rationals. Also suppose now that I have an irreducible quadratic f(x) in Q[x] with one root in K. From the problem as is, I am supposed to find such a K and f(x) for which there is only one root in K. But from the quadratic formula it seems to me that as soon as one root is in K, the other must be as well. What am I missing?
 
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Homework Equations For a quadratic f(x), the roots are given by the quadratic formula: x = -b ± √(b^2 - 4ac)/2a.The Attempt at a Solution I am still working on this problem. I am starting to think that it will be necessary to use the fact that K is a field extension of degree 2 over Q, but I'm not sure how to do that yet.
 

1. What are field extensions of Q?

Field extensions of Q refer to the extension of the field of rational numbers (Q) to a larger field, such as the field of real numbers (R) or complex numbers (C). This allows for the inclusion of new elements that cannot be expressed as fractions of integers, such as irrational numbers.

2. How are field extensions of Q represented?

Field extensions of Q are typically represented as Q(x), where x is the new element added to the field. The elements of the extended field can then be expressed as a combination of rational numbers and x, using the operations of addition, subtraction, multiplication, and division.

3. What is the degree of a field extension of Q?

The degree of a field extension of Q is the number of elements in the extended field. For example, the degree of the field extension Q(√2) is 2, since it contains the rational numbers and the square root of 2. The degree of Q(√2, √3) is 4, since it contains the rational numbers, √2, √3, and their combinations.

4. How are field extensions of Q related to algebraic and transcendental numbers?

Field extensions of Q can include both algebraic and transcendental numbers. An algebraic number is a root of a non-zero polynomial with rational coefficients, while a transcendental number cannot be expressed as a root of any polynomial with rational coefficients. For example, the field extension Q(π) includes the transcendental number π, while the field extension Q(√2) includes the algebraic number √2.

5. What are some applications of field extensions of Q?

Field extensions of Q have various applications in mathematics and science, such as in the study of algebraic geometry, number theory, and cryptography. They are also used in physics and engineering to model and solve problems involving irrational and complex numbers.

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