document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1084201>Question 1084201</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please help me to solve that the polynomial {x^2-3} is irreducible over the field of rational numbers </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #698277 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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/cgi-bin/show-face.mpl?user=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=698277\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We use the fact that f(x) is irreducible over the field of rational numbers(Q) if and only if f(x+a) is irreducible for any a an element of Q.<BR>\n" );
document.write( ":<BR>\n" );
document.write( "We prove that the polynomial f(x+1) is irreducible<BR>\n" );
document.write( ":<BR>\n" );
document.write( "We have<BR>\n" );
document.write( ":<BR>\n" );
document.write( "(x+1)^2 - 3 = x^2 +2x + 1 - 3 = x^2 +2x -2<BR>\n" );
document.write( ":<BR>\n" );
document.write( "since the nonzero coefficient of highest degree is 1(x^2 term), f(x+1) is monic <BR>\n" );
document.write( ":<BR>\n" );
document.write( "all the non-leading coefficients are divisible by the prime number 2(the 2x term)<BR>\n" );
document.write( ":<BR>\n" );
document.write( "******************************************************************************<BR>\n" );
document.write( "Eisenstein's Criterion\r<BR>\n" );
document.write( "\n" );
document.write( "Suppose we have the following polynomial with integer coefficients.\r<BR>\n" );
document.write( "\n" );
document.write( "P(x) = a(n)x^n + a(n&#8722;1)x(n&#8722;1) +...+ &#8943; + a(1)x + a(0)\r<BR>\n" );
document.write( "\n" );
document.write( "If there exists a prime number p such that the following three conditions all apply:\r<BR>\n" );
document.write( "\n" );
document.write( "    p divides each a(i) for i not equal to n,<BR>\n" );
document.write( "    p does not divide a(n), and<BR>\n" );
document.write( "    p^2 does not divide a(0),\r<BR>\n" );
document.write( "\n" );
document.write( "then Q is irreducible over the rational numbers<BR>\n" );
document.write( "****************************************************************************<BR>\n" );
document.write( "Since the constant term is not divisible by 2^2, Eisenstein’s criterion implies that the polynomial f(x+1) is irreducible over Q<BR>\n" );
document.write( ":<BR>\n" );
document.write( "Therefore by the fact stated above, the polynomial f(x) is also irreducible over Q.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );