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YCor
YCor
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Let $K/Q$ be an algebraic extension of the rational numbers. J. Robinson has proved that exists a formula $\psi$ such that when applied$\psi(x)$ in the language of rings which,applied to the rational numbers, defines the seperated set is percisely the ring integers (making the theory of Q$\mathbb{Q}$ undecidable, due to Godel's theorem). However, due to Tarski's results, the theory of the real field is complete and therefore cannot define the ring of integers $\mathbb{Z}$ is not definable in $\mathbb{R}$.

ILet $K/\mathbb{Q}$ be an algebraic extension of the rational numbers. I will ask two related questions:

  1. Without any further assumptions on the extension, are the integersis $\mathbb{Z}$ necessarily a definable setsubset in $<K,+,\times,0,1>$$\langle K,+,\times,0,1\rangle$?
  2. What purely algebraic properties of the extension might yield a more conclusive answer to the above questions? Specifically: seperabilityseparability, dimension, normality, finiteness, simplicity (maybe even Galois group structure). Moreover, is there a deeper side to this? Do there exist two extensions $K_{1}/Q,K_{2}/Q$$K_{1}/\mathbb{Q},K_{2}/\mathbb{Q}$ that all satisfy exactly the exacy same algebraic properties denoted above, yet in one of them $\mathbb Z$ is a definable setsubset, and in the other it is not?

These are not questions on which I have pondered for long, but I am curious whether there exist someany interesting examples or logical tools needed in odrderorder to tackle these questions.

Let $K/Q$ be an algebraic extension of the rational numbers. J. Robinson has proved that exists a formula $\psi$ such that when applied to the rational numbers, the seperated set is percisely the ring integers (making the theory of Q undecidable, due to Godel's theorem). However, due to Tarski's results, the theory of the real field is complete and therefore cannot define the integers.

I will ask two related questions:

  1. Without any further assumptions on the extension, are the integers necessarily a definable set in $<K,+,\times,0,1>$?
  2. What purely algebraic properties of the extension might yield a more conclusive answer to the above questions? Specifically: seperability, dimension, normality, finiteness, simplicity (maybe even Galois group structure). Moreover, is there a deeper side to this? Do there exist two extensions $K_{1}/Q,K_{2}/Q$ that all satisfy the exacy same algebraic properties denoted above, yet in one of them $\mathbb Z$ is a definable set, and in the other it is not?

These are not questions on which I have pondered for long, but I am curious whether exist some interesting examples or logical tools needed in odrder to tackle these questions.

J. Robinson has proved that exists a formula $\psi(x)$ in the language of rings which,applied to the rational numbers, defines the the ring integers (making the theory of $\mathbb{Q}$ undecidable, due to Godel's theorem). However, due to Tarski's results, the theory of the real field is complete and therefore the ring of integers $\mathbb{Z}$ is not definable in $\mathbb{R}$.

Let $K/\mathbb{Q}$ be an algebraic extension of the rational numbers. I will ask two related questions:

  1. Without any further assumptions on the extension, is $\mathbb{Z}$ necessarily a definable subset in $\langle K,+,\times,0,1\rangle$?
  2. What purely algebraic properties of the extension might yield a more conclusive answer to the above questions? Specifically: separability, dimension, normality, finiteness, simplicity (maybe even Galois group structure). Moreover, is there a deeper side to this? Do there exist two extensions $K_{1}/\mathbb{Q},K_{2}/\mathbb{Q}$ that satisfy exactly the same algebraic properties denoted above, yet in one of them $\mathbb Z$ is a definable subset, and in the other it is not?

These are not questions on which I have pondered for long, but I am curious whether there exist any interesting examples or logical tools needed in order to tackle these questions.

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George Peterzil
George Peterzil
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Definability of the ring of integer in algebraic extensions of $\mathbb Q$

Let $K/Q$ be an algebraic extension of the rational numbers. J. Robinson has proved that exists a formula $\psi$ such that when applied to the rational numbers, the seperated set is percisely the ring integers (making the theory of Q undecidable, due to Godel's theorem). However, due to Tarski's results, the theory of the real field is complete and therefore cannot define the integers.

I will ask two related questions:

  1. Without any further assumptions on the extension, are the integers necessarily a definable set in $<K,+,\times,0,1>$?
  2. What purely algebraic properties of the extension might yield a more conclusive answer to the above questions? Specifically: seperability, dimension, normality, finiteness, simplicity (maybe even Galois group structure). Moreover, is there a deeper side to this? Do there exist two extensions $K_{1}/Q,K_{2}/Q$ that all satisfy the exacy same algebraic properties denoted above, yet in one of them $\mathbb Z$ is a definable set, and in the other it is not?

These are not questions on which I have pondered for long, but I am curious whether exist some interesting examples or logical tools needed in odrder to tackle these questions.