I am a student of physics but I am doing PhD in mathematical Physics so I have to also work on MSc math topics ..and your channel is best I have found. Very good..
I'm struggling with a counter example to both approaches: x^2+4x+16. By Eisenstein, p=2 (or using 2^2), it should be reducible. If we use Z_3 where p=3 x^2+x+1 shows it should be reducible when x=1 a factor (a 0 in Z_3). The original roots are complex to boot. If I introduce x=x+1, then it changes to x^2+6x+21. Then using p=3 satisfies Eisenstein. Is there a way to know when to not trust the false positive given by Eisenstein and the Zp test and/or which element to use to test?
eisenstein's criterion isnt if and only if statement, matlab if we take a p, and it doesnt satisfy eisenstein, that doesnt mean that the polynomial is reducible; the result is inconclusive, and that means we have to use a different method to check irreducibility. for x2+4x+16, if we take p=5, there are no roots in Z5, which means x2+4x+1 (the reduced polynomial in Z5) is irreducible in Z5, and hence x2+4x+16 is irreducible in Z. hope that clears it up
Good effort brother i like your content
Especially for field theory
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Really, explain in easy way.......,👍
Ur efforts r commendable
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Nice explanation
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Very good explanation
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I am a student of physics but I am doing PhD in mathematical Physics so I have to also work on MSc math topics ..and your channel is best I have found. Very good..
Thanks
Thank you so much
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Sir, if P is not prime then this result is true or not .
I'm struggling with a counter example to both approaches: x^2+4x+16. By Eisenstein, p=2 (or using 2^2), it should be reducible. If we use Z_3 where p=3 x^2+x+1 shows it should be reducible when x=1 a factor (a 0 in Z_3). The original roots are complex to boot. If I introduce x=x+1, then it changes to x^2+6x+21. Then using p=3 satisfies Eisenstein. Is there a way to know when to not trust the false positive given by Eisenstein and the Zp test and/or which element to use to test?
eisenstein's criterion isnt if and only if statement, matlab if we take a p, and it doesnt satisfy eisenstein, that doesnt mean that the polynomial is reducible; the result is inconclusive, and that means we have to use a different method to check irreducibility. for x2+4x+16, if we take p=5, there are no roots in Z5, which means x2+4x+1 (the reduced polynomial in Z5) is irreducible in Z5, and hence x2+4x+16 is irreducible in Z. hope that clears it up
just realised you're not indian lol, matlab means "as in" in this context
over Z[x] pr bhi yahi proof hoga?
Yes absolutely right because Z[x] is contained in Q[x]....
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@@akshitsharma5267 same proof will work just replace Q[x] by Z[x]...
Very helpful vedio
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@@PathFindersAcademy vse jo mkoo cahiue tha vo ni mila video me..can uh plz solve my query