I have been reading Miles Reid Undergraduate Commutative Algebra and in chapter 4 he talks about a crucial difference between algebraic extension and integral extension (see the picture below).enter image description here

Now I can't see what crucial difference he is talking about, the only difference I find is in the definitions (that algebraic extensions are over fields and integral over rings and in the case of algebraic you can use any polynomial over the smaller field and in integral you use only monic polynomials over the smaller ring). I don't find there is a major difference in both extensions except we replace a field by a ring and how does that notion leads to ring of integers of a number field. (I don't know any algebraic number theory.) What does the author exactly want to say?

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    $\begingroup$ $1/2$ is algebraic over $\mathbb Z$ but not integral. $\endgroup$ – Georges Elencwajg Jul 18 '14 at 10:52
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    $\begingroup$ @GeorgesElencwajg polynomial satisfied by 1/2 over $ \mathbb{Z}$ is $ (2x - 1) $, not monic? $\endgroup$ – Andrew Miller Jul 18 '14 at 10:55
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    $\begingroup$ No, it is not monic. $\endgroup$ – Georges Elencwajg Jul 18 '14 at 10:58
  • $\begingroup$ @Andrew What do you think "monic" means? $\endgroup$ – Bill Dubuque Jul 18 '14 at 12:20

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