In some high school text books I find theorems called "Irrational Conjugate Theorem" (or similar) related to irrational roots of rational polynomials. More precisely they call the number $p-\sqrt{q}$ the "conjugate" of $p+\sqrt{q}$ ($p$, $q$ rationals, $\sqrt{q}$ not rational). The statements themselves are (more or less) correct, but I find it strange to use the phrase "conjugate" in this context.

I searched on the web and I did not find any resources above high school level that uses this term. I also searched on this forum and when a question asks about irrational conjugates (there are quite a few) the answers and comments mostly start along the lines "what do you mean by conjugate?"

My question is the following. Is there a universally accepted definition of a conjugate of irrational numbers (or certain type of irrational numbers)? If yes, I would appreciate a reference to a proper resource (not a high school text or a website helping high school students with their homework).

  • 1
    $\begingroup$ It is a special case of Galois conjugate from field extension theory, when the minimal polynomial is of degree two. The conjugate is the other root of the minimal polynomial with rational coefficients that $p-\sqrt{q}$ is a root of. Complex conjugate is another special case encountered in elementary contexts. $\endgroup$
    – Conifold
    Dec 16 '19 at 10:05
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    $\begingroup$ See e.g. Swallow, Exploratory Galois Theory, p. 105. $\endgroup$
    – Conifold
    Dec 16 '19 at 10:18

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