n 10OLS OLD). E. Simple Extensions Recall the definition of F(a). It is a field
ID: 3116585 • Letter: N
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n 10OLS OLD). E. Simple Extensions Recall the definition of F(a). It is a field such that () F Fa); (ii) a E F(a); (iii) any field containing F and a contains F(a). Use this definition to prove parts 1-5, where FS K,c F, and a K: 1 F(a)-F(a + c) and F(a)-F(ca). (Assume c 0) 2 Fa2 C Fa) and Fla+ b) S Fa, b). [F(a, b) is the field containing F, a, and b, and contained in any other field containing F, a and b.] Why are the reverse inclusions not necessarily true? 3 a + c is a root of por) iff a is a root of plx+ e); ca is a root of px) iff a is a root of p(cx). 4 Let p(x) be irreducible, and let a be a root of pix + c). Then FLx) pr + c))F(a) and F)p) Fac) Conclude that FLx(px+c)Fx(px). 5 Let p(x) be irreducible, and let a be a root of p(cx). Then FLx(p(cx)a Fla) and Fx) Conclude that FIx]/(p(cx)Fx (p(x). 6 Use parts 4 and 5 to prove the following: Fca) (a) zx)21212+x +4). (b) If a is a root ofx2 - 2 and b is a root of x2 - 4x +2, then Q(a) a(b). (c) If a is a root of x2 - 2 and b is a root of x2 -, then Q(a) a(b).Explanation / Answer
3) a+c is root of p(x)
it means p(a+c)=0 it implies a is root of p(x+c)=0
conversely
a is root of p(x+c) it means
p(a+c)=0
it means a+c is root of p(x)
simillarly for f(ca)
3) ac is root of p(x)
it means p(ac)=0 it implies a is root of p(xc)=0
conversely
a is root of p(xc) it means
p(ac)=0
it means ac is root of p(x)
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