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# Abstract Algebra

From pure math to modern cryptography to organic chemistry, abstract algebra underpins much of contemporary research. Learn the theory of groups, rings, and fields.

# Abstract Algebra Warmup

What real number does not have a multiplicative inverse?

Definition. A real number $$b$$ is called the multiplicative inverse of a if

$a \times b = b \times a = 1.$

What number $$a$$ satisfies $3 + a \equiv 0 \text{ (mod 7)?}$

Definition. Addition modulo 7 is defined by adding two numbers, dividing the sum by 7, and taking the remainder.

For example, $$3 + 6 \equiv 2$$ (mod 7), since 3 + 6 = 9 and the remainder when 9 is divided by 7 is 2.

What number $$a$$ satisfies $3 \times a \equiv 1 \text{ (modulo 7)?}$

Definition. Multiplication modulo 7 is defined by multiplying two numbers, dividing the product by 7, and taking the remainder.

For example, $$3 \times 6 \equiv 4$$ (mod 7), since $$3 \times 6 = 18,$$ and the remainder when 18 is divided by 7 is 4.

For which value of $$n$$ does $5 \times a \equiv 1 \text{ (mod n)}$

have a solution?

Definition. Multiplication modulo n is defined by multiplying two numbers, dividing the product by n, and taking the remainder.

For example, take $$n = 6.$$ We have $$5 \times 5 \equiv 1$$ (mod 6), since $$5 \times 5 = 25,$$ and the remainder when 25 is divided by 6 is 1.

Does every real number have an additive inverse?

Definition. A real number $$b$$ is called the additive inverse of $$a$$ if

$a + b = b + a = 0.$

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