# Confused about lemma in Ivan Niven's Irrational Numbers about conjugate elements in a number field

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 $$\alpha_i + \beta_i$$?

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!