7.3. ADJOINING ALGEBRAIC ELEMENTS TO A FIELD 325 (c) dim K .K.˛// D deg .f / . Proof. We have shown that the ring of polynomials in ˛ with coefﬁcients in K is a ﬁeld, isomorphic to KŒxŁ=.f.x// . Therefore, K.˛/ D KŒ˛Ł Š KŒxŁ=.f.x// . This shows part (a). For any p 2 KŒxŁ , write p D qf C r , where r D0 or deg .r/ < deg .f / . Then p.˛/ D r.˛/ , since f.˛/ D0 . This means that f 1;˛;:::;˛ d ± 1 g spans K.˛/ over K . But this set is also linearly independent over K , because ˛ is not a solution to any equation of degree less than d . This shows parts (b) and (c). n Example 7.3.7. Consider f.x/ D x 2 ± 2 2 Q .x/ , which is irreducible by Eisenstein’s criterion (Proposition 6.8.4 ). The element p 2 2 R is a root of f.x/ . (The existence of this root in R is a fact of analysis .) The ﬁeld Q . p 2/ Š Q ŒxŁ=.x 2 ± 2/ consists of elements of the form a C b p 2 , with a;b 2 Q . The rule for addition in Q . p
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