Unformatted text preview: can’t occur. 8. (20 points) Let m and n be distinct squarefree integers not equal to 1. (a) Prove that Q ( √ m, √ n ) is Galois over Q with Galois group Z / 2 × Z / 2. Does this ﬁeld have any quadratic subﬁelds other than Q ( √ m ) and Q ( √ n )? (b) Suppose p ramiﬁes in both Q ( √ m ) and Q ( √ n ). What happens in K ? Find an example. (c) Suppose p splits in both of these ﬁelds. What happens in K ? Find an example. (d) Suppose p is inert in both of these ﬁelds. What happens in K ? Find an example. (e) Suppose the splitting behavior of p is diﬀerent in both of these ﬁelds. What happens in K ? Find an example. (f) ( David ZureickBrown’s favorite algebraic number theory problem ) Does there exist an irreducible polynomial which is reducible mod every prime?...
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 Spring '13
 FrankThorne
 Algebra, Number Theory, Group Theory, Prime number, Algebraic number theory, Galois group

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