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Proposition 6: Let F/k be an extension field. Let E = {α = F α is algebraic over k}. Then E is a subfield of F.

Proposition 6: Let F/k be an extension field. Let E = {α = F α is algebraic over k}. Then E is a subfield of F.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Requent explain this proposition 4 proof in simple teims. Unable to understand the proof Proposition 6: Let F/k be an extension field. Let E = {α = F α is algebraic over k}. Then E is a subfield of F. Proof: Firstly, k≤E, since k/k is algebraic. Next, let α,ße E and let n,m denote, respectively, their degrees over k. Then, by Theorem 7, [k(α): k]=n and [k(ß): k] = m. Since ẞ is algebraic over k and kk(a), ß is also algebraic over k(a). Hence, [k(a)(B): k(a)] ≤ m. Now, by Theorem 3, it follows that [k(a)(B): k] = [k(a)(ß): k(a)][k(a): k] ≤mn. So k(a)(B)/k is a finite extension, and hence an algebraic extension. Since k(a)(B) is a field containing a and B, it contains a ±ß,aß and a/ß (for B = 0). Hence all these elements are algebraic over k, and thus, lie in E. Thus, E is a subfield of F.
Transcribed Image Text:Requent explain this proposition 4 proof in simple teims. Unable to understand the proof Proposition 6: Let F/k be an extension field. Let E = {α = F α is algebraic over k}. Then E is a subfield of F. Proof: Firstly, k≤E, since k/k is algebraic. Next, let α,ße E and let n,m denote, respectively, their degrees over k. Then, by Theorem 7, [k(α): k]=n and [k(ß): k] = m. Since ẞ is algebraic over k and kk(a), ß is also algebraic over k(a). Hence, [k(a)(B): k(a)] ≤ m. Now, by Theorem 3, it follows that [k(a)(B): k] = [k(a)(ß): k(a)][k(a): k] ≤mn. So k(a)(B)/k is a finite extension, and hence an algebraic extension. Since k(a)(B) is a field containing a and B, it contains a ±ß,aß and a/ß (for B = 0). Hence all these elements are algebraic over k, and thus, lie in E. Thus, E is a subfield of F.
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