TITLE: How would you explain a quadratic field to a beginner?
QUESTION [7 upvotes]: How would you explain a quadratic field to a beginner? Eg. how did the subject first start? All the modern stuff they use to explain it makes it really confusing how one should think about it in more concrete terms.
Please also give a example question and answer.
For example, a college sophomore like me who knows things through calculus, high school algebra, and some modular arithmetic.

REPLY [2 votes]: I'll start with the disclaimer that I've never been a math teacher and I'd probably be awful at it.
First I'd want to know whether the beginner's interest in mathematics is more algebraic or more geometric.
By algebraic, I'm talking about solving equations like $x^2 + 3y^2 = 5$ or $x^2 - 47y^2 = 37$. An example question would be whether these equations have solutions in integers. The answer is that the first equation does not, while the latter has infinitely many (e.g., $x = 15$, $y = 2$). Surprisingly, determining the former has no solutions can make use of imaginary numbers. You can try solving equations like these by just trying a whole bunch of values, which might not be so bad in the plus case, but can easily be trying in the minus case even if there are infinitely many solutions.
For someone with more geometric interests, I would go ahead and introduce the concept of the complex plane and numbers of the form $a + b \sqrt{-d}$, where $d$ is a positive integer and $a$ and $b$ are any integers (including $0$). You can't really do greater than or less than on these numbers, but you can compare how far these numbers are from $0$. An example question would be which of $-7 + 5 \sqrt{-2}$ and $12 - \sqrt{-2}$ is closer to $0$, and what part the Pythagorean theorem plays in this.
For someone more interested in the historical aspect of it, I suppose it comes down to the romance of Fermat's last theorem, "a simple knot tied by a part-time French mathematician working alone without a computer" and not untied quite to the satisfaction of a French space captain in the age of truly ubiquitous computers.
There is also the number theoretic perspective, for questions such as what prime numbers can be solutions to an equation like $x^2 - 10y^2$. It's not for nothing that there is an entire book titled Fermat's Last Theorem and Algebraic Number Theory.