Abstract Algebra - UCLA Department of Mathematics

... performed on them. For instance, one may add and multiply integers, and one can do the same with rational numbers, real numbers, and even complex (or imaginary) numbers. Or, given two functions that input and output real numbers, we can compose them. We can add vectors, or multiply them by scalars. ...

... performed on them. For instance, one may add and multiply integers, and one can do the same with rational numbers, real numbers, and even complex (or imaginary) numbers. Or, given two functions that input and output real numbers, we can compose them. We can add vectors, or multiply them by scalars. ...

Solutions

... of 3 are additive inverses mod 32, all 16 odd congruence classes mod 32 can be written as either one of the above numbers or its negative. (b) Determine the order of 9 modulo 64. By the above list of powers, 9 has order 4 modulo 32. So lifting to 64, 9 has order divisible by 4. In fact, 94 ≡ 33 (mod ...

... of 3 are additive inverses mod 32, all 16 odd congruence classes mod 32 can be written as either one of the above numbers or its negative. (b) Determine the order of 9 modulo 64. By the above list of powers, 9 has order 4 modulo 32. So lifting to 64, 9 has order divisible by 4. In fact, 94 ≡ 33 (mod ...

Geometric Constructions from an Algebraic Perspective

... Theorem 24. Let α be a real number. Then α is constructible if and only if α belongs to the top of some square root tower over Q. Proof. (⇐)Let C be the set of constructible real numbers. C is an extension field of Q. C is a subfield of R because we have shown earlier that the constructible set C is ...

... Theorem 24. Let α be a real number. Then α is constructible if and only if α belongs to the top of some square root tower over Q. Proof. (⇐)Let C be the set of constructible real numbers. C is an extension field of Q. C is a subfield of R because we have shown earlier that the constructible set C is ...

Intro Abstract Algebra

... Note that, for example, 1 6= f1g, and ff1gg 6= f1g. That is, the set fag with sole element a is not the same thing as the item a itself. An ordered pair (x; y) is just that, a list of two things in which there is a rst thing, here x, and a second thing, here y. Two ordered pairs (x; y) and (x0 ; y0 ...

... Note that, for example, 1 6= f1g, and ff1gg 6= f1g. That is, the set fag with sole element a is not the same thing as the item a itself. An ordered pair (x; y) is just that, a list of two things in which there is a rst thing, here x, and a second thing, here y. Two ordered pairs (x; y) and (x0 ; y0 ...

Algebraic Number Theory Brian Osserman

... Kummer developed the theory of ideals of rings in part to prove the following remarkable theorem: Theorem 1.3.6. Suppose that p is an odd prime number such that p does not divide any of the numerators of B2 , B4 , . . . , Bp−3 (in particular, p = 3 is acceptable). Then the equation xp + y p = z p ha ...

... Kummer developed the theory of ideals of rings in part to prove the following remarkable theorem: Theorem 1.3.6. Suppose that p is an odd prime number such that p does not divide any of the numerators of B2 , B4 , . . . , Bp−3 (in particular, p = 3 is acceptable). Then the equation xp + y p = z p ha ...

# Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.In field theory and ring theory the notion of root of unity also applies to any ring with a multiplicative identity element. Any algebraically closed field has exactly n nth roots of unity, if n is not divisible by the characteristic of the field.