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Page 136 of Ivan Niven's "Irrational Numbers" states, "Lemma 10.4: Let $\alpha$, $\beta$ be algebraic numbers in a field $K$ of degree $h$ over the rationals. If the conjugates for $\alpha$ for $K$ are $\alpha_1, \alpha_2 \ldots, \alpha_h$", then the conjugates of $\alpha\beta$ and $\alpha + \beta$ are $\alpha_1\beta_1, \ldots, \alpha_h\beta_h$ and $\alpha_1 + \beta_1, \ldots, \alpha_h + \beta_h$."

I am confused about this statement. I learned that the conjugates are the roots of the minimal polynomial. It seems that the book later assumes(?) all $\alpha \in K$ to have exactly $h$ conjugates (or at least everything in $\mathcal{O}_K$). I thought some elements could have a minimal polynomial with degree < $h$. I was wondering if they meant conjugate element in another sense (do they mean $\sigma_i(\alpha)$ for the $h$ embeddings $K \hookrightarrow \mathbb{C}$ that fix $\mathbb{Q}$ instead?) Also, by $\alpha_1\beta_1, \ldots, \alpha_h\beta_h$ and $\alpha_1 + \beta_1, \ldots, \alpha_h + \beta_h$, do they mean $\alpha_i\beta_j$ and $\alpha_i + \beta_j$ or $\alpha_i\beta_i$ and $\alpha_i + \beta_i$?

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For your first question: algebraic elements in general can have minimial polynomials of large degree, but the algebraic elements in this degree-$h$ field extension all have minimial polynomials of degree at most $h$ (indeed the degree must divide $h$, as we see from $[K:\Bbb Q] = [K:\Bbb Q(\theta)][\Bbb Q(\theta):\Bbb Q]$). If the degree is strictly less than $h$, then some of the conjugates will be repeated (consider $\beta=1/\alpha$ and $\beta=-\alpha$ for example ... or for that matter $\alpha=1$ directly).

For your second question: in this whole context, they are referring to conjugates in the sense of $\Bbb Q$-fixing embeddings. Your intuition is right that there is a link to roots of a polynomial, but we need to be careful: the two concepts (field embeddings and roots of the minimal polynomial) coincide only when the field in question is exactly $\Bbb Q(\theta)$ where the $\theta$ is the algebraic number we're taking conjugates of.

For your last question: they really mean $\alpha_i\beta_i$ and not $\alpha_i\beta_j$. Try some examples by hand, say in $\Bbb Q[i]$—there's nothing like actual computation to help us internalize abstract statements!

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    $\begingroup$ What definition of conjugate are you using in the first of the "two concepts" (i.e. not the one involving field embeddings)? $\endgroup$
    – FShrike
    Nov 29, 2023 at 18:41

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