Math 114 - Spring 2001 - Bergman - Final

# Math 114 - Spring 2001 - Bergman - Final - 10/05/2001 FRI...

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Unformatted text preview: 10/05/2001 FRI 17:15 FAX 0434330 MOFFITT LIBRARY 001 George M. Bergman Spring 2001, Math 114 15 May, 2001 70 Evans Hall Final Examination 12:30-3230 PM 1. (20 points, 4 points each.) Complete each of the following deﬁnitions. (Do not give examples or other additional facts about the concepts deﬁned.) (a) If K is a ﬁeld and f(t), g(t) are nonzero elements of K [t], then a highest common factor of f(t) and g(t) means (b) If L:K is aﬁeld extension, then the degree of L over K (written [LzK]) means (C) If L:K is a ﬁeld extension and H is a subgroup of F(L:K), then H7 is deﬁned as (d) If L is a ﬁnitely generated extension of the ﬁeld K, then the transcendence degree of L over K (also written tr.deg.(L:K)) means (In answering this, you may assume facts and/or deﬁnitions preceding the deﬁnition of transcendence degree in either Stewart 01' my handout on the subject. However, do not refer as Stewart does to “the number deﬁned in” a certain lemma of his, without re—stating the Conditions of that lemma.) (e) If K is a ﬁeld of characteristic p i 0, then the Frobenins endomorphism of K (in Stewart’s language, the ‘Frobenius monomorphism’) is 2. (20 points; 4 points each.) For each of the items listed below, either give an example, or give a brief reason why no example exists. (If you give an example, you do not have to prove that it has the property stated.) (a) A ﬁeld extension L:K and an intermediate ﬁeld M (i.e., a ﬁeld with K g M g L) such that [L:K] = 3 and [MzK] = 2. (b) A ﬁeld extension LzK such that the characteristic of K and the characteristic of L are not equal. (c) A ﬁnite inseparable ﬁeld extension. ((1) An irreducible quintic polynomial (i.e., polynomial of degree 5) f(t)e®[t] which is solvable in radicals. (e) A polynomial f(t)E<D [t] which has noncommutative Galois group over (D, but commutative Galois group over ©(w), where a) is a primitive cube root of unity. 10/05/2001 FRI 17:15 FAX 6434330 MOFFITT LIBRARY 002 3. (’7 points) Suppose L2K is an algebraic extension, 1' an element of F(L:K), and J: an element of L. Show that x and 7(x) have the same minimal polynomial over K. 4. (15 points) Let p be a prime number and n a positive integer. Show that the irreducible factors of the polynomial tpn— telp [t] are precisely all irreducible polynomials fe ZPII] whose degrees are divisors of n. (Hint: Consider the extension of 22p generated by a zero of an irreducible polynomial f. You may use the fact, proved in homework, that the subﬁelds of GF(p”) are precisely the ﬁelds GF(pm) for which ﬁt divides n.) 5. (20 points) Recall that a ﬁeld L is said to be algebraically closed if every polynomial over L of degree > 0 splits in L. (a) (8 points) Suppose L is an algebraically closed ﬁeld, and [C any subﬁeld of L. Show that if E is a ﬁnite algebraic extension of K, then E can be mapped into L by a Kumonomorphism. (Hint: If E is normal over K, you can use uniqueness of spiitting ﬁelds.) (b) (8 points) Show that if E is a proper algebraic extension of the ﬁeld {R of real numbers (i.e., if E is algebraic over {R and not equal to IR), then E is isomorphic over [R (i.e., isomorphic by an lesomorphism) to C. (Suggestion: ﬁrst do this in the case where E is ﬁnite over IR, with the help of part (a), then deduce that the case where E is inﬁnite over [R cannot occur.) (c) (4 points) Give an example of a proper ﬁnitely generated extension E: {R which is not isomorphic over [R to C, and give a reason why it is not. 6, (18 points) (a) (9 points) Suppose L is a ﬁeld of characteristic not equal to 2, in which the polynomial :4 + 1 splits. Show that the subgroup of the multiplicative group of L generated by the zeros of that polynomial is cyclic of order 8. If we let a: be a generator of that group, which elements of (0 g i < 8) are the zeros of t4 + 1 ? (b) (9 points) Suppose K is a ﬁeld of characteristic not cqual to 2, in which the 4 polynomial :4 + l is irreducible. Determine the Galois group of r + 1 over K precisely, writing out its multiplication table. (In writing this table, specify the meanings of the symbols you use for the group elements. Your discussion should show why the Galois group must have this table, though you don’t have to show all arithmetic calculations. Hint: If or is one zero of the given polynomial, part (a) shows what elements a member of the Galois group can, take a to.) ...
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## This test prep was uploaded on 04/05/2008 for the course MATH 114 taught by Professor Borcherds during the Spring '03 term at Berkeley.

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