document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199314>Question 1199314</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How to see that a complete subfield F in Q_p with absolute value | |_p, is actually Q_p itself? \r<BR>\n" );
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document.write( "We have one inclusion: F\subset Q_p.<BR>\n" );
document.write( "Trying to show that Q_p\subset F. Q_p is complete with respect to | |_p. take an element x in F\subset Q_p, so there exists a cauchy sequence x_n in Q_p such that x_n—>x.<BR>\n" );
document.write( "But F id also complete so there exists y_n in F such that <BR>\n" );
document.write( "y_n—>x, but then x_n=y_n, so can we say that Q_p\subset F and we're done? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848220 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848220\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Key Idea**\r<BR>\n" );
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document.write( "* The crux of the argument lies in the uniqueness of limits in complete metric spaces. \r<BR>\n" );
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document.write( "**2. Proof Outline**\r<BR>\n" );
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document.write( "* **Assume:** We have a complete subfield F of Q_p. This means F is a field itself, and it's complete with respect to the same p-adic absolute value |.|_p as Q_p.\r<BR>\n" );
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document.write( "* **Show Q_p ⊆ F:**<BR>\n" );
document.write( "    * Take any x ∈ Q_p. <BR>\n" );
document.write( "    * Since Q_p is complete, there exists a Cauchy sequence (x_n) in Q ⊆ F such that x_n → x in Q_p. <BR>\n" );
document.write( "    * Since F is a subfield of Q_p, it contains all rational numbers (Q). Therefore, (x_n) is also a Cauchy sequence in F.<BR>\n" );
document.write( "    * Since F is complete, this Cauchy sequence (x_n) must converge to a limit y in F.<BR>\n" );
document.write( "    * **Uniqueness of Limits:** In any metric space (and hence in Q_p and F), limits are unique. Since x_n → x in Q_p and x_n → y in F, we must have x = y.<BR>\n" );
document.write( "    * This shows that for any x ∈ Q_p, there exists a corresponding y ∈ F (namely, y = x). <BR>\n" );
document.write( "    * Therefore, Q_p ⊆ F.\r<BR>\n" );
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document.write( "* **Conclusion:**\r<BR>\n" );
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document.write( "    * We started with F ⊆ Q_p and proved Q_p ⊆ F. <BR>\n" );
document.write( "    * Combining these, we conclude that F = Q_p.\r<BR>\n" );
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document.write( "**In essence:**\r<BR>\n" );
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document.write( "* The completeness of both F and Q_p, along with the uniqueness of limits in complete metric spaces, forces any complete subfield of Q_p to be equal to Q_p itself.\r<BR>\n" );
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document.write( "**Note:** This proof relies heavily on the uniqueness of limits in complete metric spaces. \r<BR>\n" );
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document.write( "Let me know if you have any other questions or would like to explore related concepts!<BR>\n" );
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