I always liked Edward Burger's A Tail of Two Palindromes. It begins as follows:

Upon a preliminary perusal, this parable may appear to be about pairs of palindromes, periods, and pitiful alliteration. In actuality, however, it is the story of a real quadratic irrational number $\alpha$ and its long-lost younger sibling, its algebraic conjugate $\tilde{\alpha}$ ($\alpha > \tilde{\alpha}$). How in the dickens are all these notions connected? We begin at the beginning...

Although the conjugates $\alpha$ and $\tilde{\alpha}$ are not identical twins, unlike the two zeros of $(x - 3)^2$, they do share a common family history: they each were born of the same irreducible parent polynomial having rational coefficients,

$$P_{\alpha}(x) = P_{\tilde{\alpha}}(x) = (x - \alpha)(x - \tilde{\alpha}) = x^2 - \text{Trace}(\alpha)x + \text{Norm}(\alpha),$$

where $\text{Trace}(\alpha) = \alpha + \tilde{\alpha}$ and $\text{Norm}(\alpha) = \alpha \tilde{\alpha}$. Perhaps not surprisingly, some conjugate pairs exhibit similar personalities. But how similar can they be? And how can we detect those similarities simply by looking at $\alpha$? As we will discover as our tale unfolds, the answer - foreshadowed in the title - is encoded in what can be described as the number theoretic analogue of the DNA-sequence for $\alpha$. However, before delving into $\alpha$'s genes, we first motivate our results by weaving a lattice of algebra.

I always liked Edward Burger's A Tail of Two Palindromes. It begins as follows:

Upon a preliminary perusal, this parable may appear to be about pairs of palindromes, periods, and pitiful alliteration. In actuality, however, it is the story of a real quadratic irrational number $\alpha$ and its long-lost younger sibling, its algebraic conjugate $\tilde{\alpha}$ ($\alpha > \tilde{\alpha}$). How in the dickens are all these notions connected? We begin at the beginning...

Although the conjugates $\alpha$ and $\tilde{\alpha}$ are not identical twins, unlike the two zeros of $(x - 3)^2$, they do share a common family history: they each were born of the same irreducible parent polynomial having rational coefficients,

$$P_{\alpha}(x) = P_{\tilde{\alpha}}(x) = (x - \alpha)(x - \tilde{\alpha}) = x^2 - \text{Trace}(\alpha)x + \text{Norm}(\alpha),$$

where $\text{Trace}(\alpha) = \alpha + \tilde{\alpha}$ and $\text{Norm}(\alpha) = \alpha \tilde{\alpha}$. Perhaps not surprisingly, some conjugate pairs exhibit similar personalities. But how similar can they be? And how can we detect those similarities simply by looking at $\alpha$? As we will discover as our tale unfolds, the answer - foreshadowed in the title - is encoded in what can be described as the number theoretic analogue of the DNA-sequence for $\alpha$. However, before delving into $\alpha$'s genes, we first motivate our results by weaving a lattice of algebra.

I always liked Edward Burger's A Tail of Two Palindromes. It begins as follows:

Upon a preliminary perusal, this parable may appear to be about pairs of palindromes, periods, and pitiful alliteration. In actuality, however, it is the story of a real quadratic irrational number $\alpha$ and its long-lost younger sibling, its algebraic conjecture $\tilde{\alpha}$ ($\alpha > \tilde{\alpha}$). How in the dickens are all these notions connected? We begin at the beginning...

Although the conjugates $\alpha$ and $\tilde{\alpha}$ are not identical twins, unlike the two zeros of $(x - 3)^2$, they do share a common family history: they each were born of the same irreducible parent polynomial having rational coefficients,

$$P_{\alpha}(x) = P_{\tilde{\alpha}}(x) = (x - \alpha)(x - \tilde{\alpha}) = x^2 - \text{Trace}(\alpha)x + \text{Norm}(\alpha),$$

where $\text{Trace}(\alpha) = \alpha + \tilde{\alpha}$ and $\text{Norm}(\alpha) = \alpha \tilde{\alpha}$. Perhaps not surprisingly, some conjugate pairs exhibit similar personalities. But how similar can they be? And how can we detect those similarities simply by looking at $\alpha$? As we will discover as our tale unfolds, the answer - foreshadowed in the title - is encoded in what can be described as the number theoretic analogue of the DNA-sequence for $\alpha$. However, before delving into $\alpha$'s genes, we first motivate our results by weaving a lattice of algebra.

I always liked Edward Burger's A Tail of Two Palindromes. It begins as follows:

Upon a preliminary perusal, this parable may appear to be about pairs of palindromes, periods, and pitiful alliteration. In actuality, however, it is the story of a real quadratic irrational number $\alpha$ and its long-lost younger sibling, its algebraic conjugate $\tilde{\alpha}$ ($\alpha > \tilde{\alpha}$). How in the dickens are all these notions connected? We begin at the beginning...

Although the conjugates $\alpha$ and $\tilde{\alpha}$ are not identical twins, unlike the two zeros of $(x - 3)^2$, they do share a common family history: they each were born of the same irreducible parent polynomial having rational coefficients,

$$P_{\alpha}(x) = P_{\tilde{\alpha}}(x) = (x - \alpha)(x - \tilde{\alpha}) = x^2 - \text{Trace}(\alpha)x + \text{Norm}(\alpha),$$

where $\text{Trace}(\alpha) = \alpha + \tilde{\alpha}$ and $\text{Norm}(\alpha) = \alpha \tilde{\alpha}$. Perhaps not surprisingly, some conjugate pairs exhibit similar personalities. But how similar can they be? And how can we detect those similarities simply by looking at $\alpha$? As we will discover as our tale unfolds, the answer - foreshadowed in the title - is encoded in what can be described as the number theoretic analogue of the DNA-sequence for $\alpha$. However, before delving into $\alpha$'s genes, we first motivate our results by weaving a lattice of algebra.