Question

Abstract Algebra: Let
K = Q(V2,0)
. It has been shown already that K is the splitting field over $\mathbb{Q}$, and the following isomorphisms are of onto a subfield $\overline{\mathbb{Q}}$ as extensions of the automorphism $\iota_\mathbb{Q}:\mathbb{Q}\rightarrow\mathbb{Q}$ , and also the elements of :

;

;

;

.

We also proved previously that is separable over $\mathbb{Q}$. Based on all of those outcomes, find all subgroups of and their corresponding fixed fields as the intermediate fields between $\mathbb{Q}$ and , and complete the subgroup and subfield diagrams for them. Finally, establish the one-to-one correspondence between the subgroups and fixed fields that shows Galois Theory's main theorem.

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Answer 1

k=8 (52, i) Elements of G(KIB) are 4,: 8(521) ► 8 (52,6) 02:8 (52, i) 8(-52 ,i) 03 : 8 (32,1) + 8 (32,-) &4: B (52, i) 8 (-62,-1) So, G(K/B) = { 0,92,93,945 & 4. = Id now, sub-group of G/8 are se afla {i} , {91,929, {91,83} and {P1, 949 Fixed field of soig is & (52, il 1 1 ŞQ1,824 is scº) $,193 0 (52) " " " $ 21,94% . fixed field so, there is one to one correspondence between of a K and subgroup of G(K/6)