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I'm going through Smart and Vercauteren's paper "Fully Homomorphic SIMD operations" and had a question about some notation used in the paper.

In section 2 of the above it is stated that for each monic polynomial $F(X) \in \mathbb{F}_2[X]$ of degree $N$ that splits into $r$ distinct irreducible factors of degree $d = N/r$ viz. $F(X)=\prod_{i = 1}^{r}F_i(X)$, the polynomial $F(X)$ defines the field $\mathbb{K} = \mathbb{Q}(\theta) = \mathbb{Q}[X]/(F)$ where $\theta$ is a fixed root in the algebraic closure of the base field.

Questions:

  1. Should it not be stated that $\mathbb{Q}(\theta) \cong \mathbb{Q}[X]/(F)$ rather than $\mathbb{Q}(\theta) = \mathbb{Q}[X]/(F)$ ?

  2. I assume that $\theta$ is just a root of $F(X)$ in the extension $\mathbb{Q}/\mathbb{F}_2$?

  3. In the expression $\mathbb{Q}[X]/(F)$, is what is being referred to the
    quotient of $\mathbb{Q}[X]$ by the ideal generated by $F(X)$, i.e. $\mathbb{Q}[X]/(F(X))$?

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Regarding question (2), I would restate it as: Is what is meant by $\mathbb{Q}(\theta) = \mathbb{Q}[X]/(F)$ the field $\mathbb{Q}(\theta)$ created by adjoining $\theta$, a root of $F(X)$ in the algebraic closure of $\mathbb{F}_2$, to $\mathbb{Q}$? –  Rohit Khera Apr 4 at 22:37

1 Answer 1

  1. Yes, technically this is an isomorphism, so $\cong$ should be used. The isomorphism is question identifies $\theta$ with $X$.

  2. Not quite (for the original question or your restatement). The notation $\mathbb{Q}(\theta)$ means adjoining $\theta$, a root of $F(X)$ in the algebraic closure of $\mathbb{Q}$ (namely, $\mathbb{C}$), to $\mathbb{Q}$. The finite field $\mathbb{F}_2$ has nothing to do with it. (And I think $\mathbb{Q}/\mathbb{F}_2$ is meaningless, since there is no field embedding of $\mathbb{F}_2$ into $\mathbb{Q}$.)

  3. Yes. Such a structure is more commonly known as a number field (i.e., a finite dimensional field extension of the rationals).

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